# A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm

Authors: Wilhelm P. Freire; Afonso C. C. Lemonge; Tales L. Fonseca; Hernando J. R. Franco
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

The Interior Epigraph Directions (IED) method for solving constrained nonsmooth and nonconvex optimization problem via Generalized Augmented Lagrangian Duality considers the dual problem induced by a Generalized Augmented Lagrangian Duality scheme and obtains the primal solution by generating a sequence of iterates in the interior of the epigraph of the dual function. In this approach, the value of the dual function at some point in the dual space is given by minimizing the Lagrangian. The first version of the IED method uses the Matlab routine fminsearch for this minimization. The second version uses NFDNA, a tailored algorithm for unconstrained, nonsmooth and nonconvex problems. However, the results obtained with fminsearch and NFDNA were not satisfactory. The current version of the IED method, presented in this work, employs a Genetic Algorithm, which is free of any strategy to handle the constraints, a difficult task when a metaheuristic, such as GA, is applied alone to solve constrained optimization problems. Two sets of constrained optimization problems from mathematics and mechanical engineering were solved and compared with literature. It is shown that the proposed hybrid algorithm is able to solve problems where fminsearch and NFDNA fail.


---

## Body

## 1. Introduction

We present a new version of the Interior Epigraph Directions Method published by Burachik et al. [1] for solving constrained nonsmooth and nonconvex optimization problems of the following kind: Pminimizefxoverallx∈Xsatisfyinggx=0,where X⊂Rn is compact and the functions f:Rn→R and g:Rn→Rm are continuous. Inequality constraints can be incorporated into P by using the operator a+=max⁡a,0, a∈R.The IED method uses a Lagrangian duality technique, where each constraint function is appended to the objective function via a multiplier, or dual variable, to form the classical Lagrangian. In a duality scheme, instead of problemP, the problem solved is the one where the dual function is maximized. In this approach, the value of the dual function at a given point in the dual space is computed by minimizing the Lagrangian which is an unconstrained problem, and therefore unconstrained methods can be used.It turns out that the optimal value of the dual problem associated with the classical Lagrangian is in general not the same as that of the primal problemP, if P is nonconvex. The difference between the two optimal values is called duality gap.Zero duality gap and saddle-point properties without convexity assumptions are known to hold for the following family of Lagrangian function [2, Remark 2.1].(1)Lx,u,c=fx+cσgx+Au,gx,where x∈Rn, u∈Rm, c∈R+=[0,∞), A∈Rm×m is a real symmetric matrix, and σ:Rm→R is a continuous function such that σ(x)>0, ∀x∈Rm∖{0} and σ(0)=0.The dual problem generated by such Lagrangians as in (1) is a (nondifferentiable) convex problem, which is usually solved by nonsmooth convex optimization techniques, such as subgradient methods and their extensions [2–8].The IED method uses the family of Lagrangians (1). These Lagrangians induce dual variables (u,c)∈Rm×R+, where u is the classical multiplier associated with the linear term in the Lagrangian and c≥0 is the penalty multiplier associated with the augmenting term.A duality approach based on the LagrangianL for solving P can be described as follows. Given a current dual iterate (uk,ck), a primal iterate is computed through the rule(2)xk∈arg⁡minx∈X⁡Lx,uk,ck,where L is as in (1).The iteratexk obtained in (2) can be used for computing the search direction which leads to the next dual iterate (uk+1,ck+1). Indeed, xk is used to obtain a deflection of the subgradient direction which in turn is utilized to improve the dual values. This is the idea behind the methods studied in [2–8]. These methods are referred to as deflected subgradient (DSG) methods.The IED method [1] combines the Nonsmooth Feasible Directions Algorithm (NFDA) for solving unconstrained nonsmooth convex problems, firstly presented in [9] and further studied in [10], with a certain DSG method.Roughly speaking, the IED method works as follows: starting at a point in the interior of the epigraph of the dual function, a deflected subgradient direction is used to define a linear approximation to the epigraph. An auxiliary point is obtained by solving the optimality conditions of a resulting linear problem. If the auxiliary point belongs to the interior of the epigraph, the IED step is declared serious and the auxiliary point becomes the next iterate, from which the process is repeated. If the auxiliary point does not belong to the interior of the epigraph, the IED step is declared null and a suitable version of the DSG step is applied from the original point. Thanks to the properties of the DSG method, this step stays in the interior of the epigraph. The new point takes the place of the original point, from which the process is repeated, until a serious step is performed. If an infinite number of null steps occurs, the method converges to a primal solution, as a consequence of the convergence properties of the DSG.An important aspect of the method is the minimization of the Lagrangian function. Although the solver used for this task does not affect the convergence of the IED method, as proved in [1], solving this problem effectively is fundamental for the success of the methods based on Lagrangian duality.The first version of the IED method [1] employs the Matlab routine fminsearch for the Lagrangian minimization. The second version of the IED method [15] uses an algorithm called Nonsmooth Feasible Directions Nonconvex Algorithm (NFDNA) [16, 17] for solving nonsmooth and nonconvex unconstrained optimization problems which correspond to our problem (2). Although the previous versions of IED were able to solve many problems, they failed some test problems and did not find the desired solutions as one can observe through some numerical experiments presented in this paper and found in [1, 15]. We do not know why fminsearch and NFDNA fail. For fminsearch is a routine of Matlab, we think that it does not make sense trying to fix or change it. The NFDNA performance needs deeper investigation. In this work, we have used a Genetic Algorithm (GA) for minimizing the Lagrangian function. With this modification, the method gained robustness and was able to solve all the test problems analyzed here. The hybridization of the IED method and the Genetic Algorithm is the main purpose of this work and can be considered innovative. Also, it is important to notice that the GA is free of any strategy to handle the constraints, a difficult task when a metaheuristic, such as the GA, is applied alone to solve constrained optimization problems [18].We have solved twenty problems and compared the performance of the current version of the IED method with the performance of the first version. We also solved four real-world mechanical engineering optimization problems with the current version of the IED.This paper is organized in 8 sections. In the next section, we state the primal and the dual problems and recall how to find a subgradient of the dual function. In Section3, we show how the search direction used by IED is obtained. In Section 4, we present NFDA, an algorithm for convex optimization problems that inspired the IED method. The IED method is then presented in Section 5. In Section 6, we present a brief description of the main characteristics of the Genetic Algorithm and its basic features adopted in this paper. Numerical results and comparisons are given in Section 7. Finally, conclusions and suggestions for future work are given in Section 8.


## 2. The Problem in Study

The problems we are interested in are of the same type of the primal problemP.The augmented Lagrangian functionL:Rn×Rm×R+→R associated with P is defined as in (1). The corresponding concave dual function H:Rm×R+→R is defined as(3)Hu,c=minx∈X⁡fx+cσgx+Au,gx.The dual problem is given byDmaxu,c∈Rm×R+⁡Hu,cand, since the dual function H is concave, D is equivalent to the convex problemD∗minu,c∈Rm×R+-Hu,c.For convenience, we introduce the set(4)Xu,c=arg⁡minx∈X⁡fx+cσgx+Au,gxwhose elements are obtained by the GA used in this work.Figure8 shows a flowchart that summarizes how such minimization is done.For simplicityD∗ will be our dual problem and we will refer to q≔-H as the dual function.We recall now an important result that shows how, after solving problem (4), we can easily obtain a subgradient s∈∂q(u,c).Theorem 1 (see [2], Lemma 2.1(a)).
Given the dual functionq and a pair v=(u,c)∈Rm×R+, take x∈X(v). Then s=(-Ag(x),-σ(g(x)))∈∂q(v).


## 3. The IED Search Direction

In this section we describe how the IED search direction is obtained.The methodology employed here was firstly used by an algorithm called Feasible Directions Interior Point Algorithm (FDIPA) [19] for smooth nonlinear problems.Let us consider the inequality constrained optimization problemP3minimizefx,subject  togx≤0,where f:Rn→R and g:Rn→Rm are continuously differentiable.Letx∗ be a regular point of problem P3. The Karush-Kuhn-Tucker (KKT) first order necessary optimality conditions are expressed as follows: If x∗ is a local minimum of P3 then there exists λ∗∈Rm such that(5)∇fx∗+∇gx∗λ∗=0(6)Gx∗λ∗=0(7)λ∗≥0(8)gx∗≤0,where G(x) is a diagonal matrix with Gii(x)≡gi(x).The pointx such that g(x)≤0 is called a “Primal Feasible Point” and λ≥0 a “Dual Feasible Point”. Given an initial feasible pair (x0,λ0), KKT points are found by solving iteratively the nonlinear system of equations (5) and (6) in (x,λ), in such a way that all the iterates are primal and dual feasible.A Newton-like iteration to solve the nonlinear system of equations (5) and (6) can be stated as(9)Bk∇gxkΛk∇gxktGxkx-xkλ-λk=-∇fxk+∇gxkλkGxkλk,where (xk,λk) is the current point of the iteration k, Λ is a diagonal matrix with Λii≡λi, and Bk≡∇2f(xk)+∑i=1mλik∇2gi(xk) is the Hessian of the Lagrangian function L(x,λ)=f(x)+λTg(x) or some quasi-Newton approximation. However, Bk must be symmetric and positive definite in order to ensure convergence.Callingd=x-xk, the following system can be written from (9):(10)Bkd+∇gxkλ=-∇fxk(11)Λk∇gxkTd+Gxkλ=0.The solution(d1k,λ1k) of the system (10)-(11) provides a descent direction d1k as proved in [19]. However, d1k can not be employed as a search direction, since it is not necessarily a feasible direction because, in the case when gi(xk)=0, it follows from (11) that ∇gi(xk)Td1k=0.To avoid this drawback, a new system with unknownsd and λ¯ is defined by adding a negative matrix -ρkΛk, with ρk>0, to the right side of (11),(12)Bkd+∇gxkλ¯=-∇fxk(13)Λk∇gxkTd+Gxkλ¯=-ρkΛk.Now, (13) is equivalent to λik∇gi(xk)Tdk+gi(xk)λ¯k=-ρkλik, i=1,2,…,m. Consequently, if gi(xk)=0, the latest equation is reduced to ∇gi(xk)dk=-ρk<0, which means that dk is a feasible direction.The addition of a negative term-ρkλk produces a proportional deflexion of dk into the feasible region, as one can see in Figure 1. The problem now is that dk might not be a descent direction for the function f. However, we can keep such property if ρk is properly chosen. Notice that the direction dk can be obtained by solving two systems with the same matrix(14)Bkd1+∇gxkλ1=-∇fxk(15)Λk∇gxkTd1+Gxkλ1=0,(16)Bkd2+∇gxkλ2=0(17)Λk∇gxkTd2+Gxkλ2=-Λkand setting dk=d1k+ρkd2k, which gives (dk)T∇f(xk) = (d1k)T∇f(xk) + ρk(d2k)T∇f(xk). In the case where (d2k)T∇f(xk)≤0 we have (dk)T∇f(xk)<0, ∀ρk>0.Figure 1
The search direction of FDIPA.Thus,ρk must be well chosen if (d2k)T∇f(xk)>0. In this case, imposing (dk)T∇f(xk)≤ξ(d1k)T∇f(xk), where ξ∈(0,1), implies (d1k)T∇f(xk) + ρk(d2k)T∇f(xk) ≤ ξ(d1k)T∇f(xk) which leads to (18)ρk≤ξ-1d1kT∇fxkd2kT∇fxk.Thus, with ρk chosen as described above, we have that dk=d1k+ρkd2k is a feasible descent direction.The ideas just described are sufficient for us to understand how the search direction used by the IED method is computed.For more on FDIPA, its features, and convergence, see [19].


## 4. NFDA for Convex Problems

The Nonsmooth Feasible Directions Algorithm (NFDA), firstly presented by Freire [9], has been devised for solving unconstrained nonsmooth convex problems of the following kind:P4.1minFxx∈Rn,where F:Rn→R is a convex, not necessarily differentiable function.ProblemP4.1 can be replaced with the equivalent constrained problemP4.2minfx,z=zsubject  toFx≤zx,z∈Rn×R.It is assumed that one arbitrary subgradient s∈∂F(x) can be computed at any point x∈Rn.NFDA starts at a point(x0,z0) in the interior of the epigraph of the function F. At a point (xk,F(xk)), the method determines a supporting hyperplane to the epigraph of the function F(19)hkx=Fxk+sk,x-xk,where sk∈∂F(xk) and defines an auxiliary constrained linear problem employing all the supporting hyperplanes computed so far, as follows:P4.3minfx,z=zsubject  togkx,z≤0x,z∈Rn×R,where gk=(g1k,…,gmk):Rn+1→Rm is a vector function with gik:Rn+1→R given by gik(x,z)=hik(x)-z.ProblemP4.3 is not solved. Indeed, it might not have a solution. NFDA uses its structure, which is similar to the structure of the problem P3, to obtain a search direction by solving two systems and choosing the parameter ρ in the same way FDIPA does.At the iterate(xk,zk) and having the search direction dk at hand, a step tk is obtained by the rule (20)tk=min⁡tmax⁡,T,where tmax⁡=max⁡t∣gikxk,zk+tdk≤0 and T>0 is a predefined parameter, since tmax⁡ is not always finite.A new point(yk,ωk)=(xk,zk)+μtkdk with μ∈(0,1) is computed. If F(yk)<ωk then (xk+1,zk+1)=(yk,ωk) and the step is calledserious, as shown in Figure 2. Otherwise, if F(yk)≥ωk then (xk+1,zk+1)=(xk,zk) and the step is callednull; see Figure 3.Figure 2
Serious step.Figure 3
Null step.In any case, a new supporting hyperplane at the point(yk,F(yk)) is added to form a new auxiliary problem like P4.3.These are the main ideas from NFDA; we need to understand the IED algorithm. For more on NFDA, see [9, 10].


## 5. The IED Method

In this section we describe the Interior Epigraph Directions (IED) method for solving constrained, nonsmooth, and nonconvex optimization problems.As we have seen in Section2, the dual problem D is equivalent to the problemD∗∗minqu,cu,c∈Rm×R+which, in its turn, is equivalent toED∗minψu,c,z=zsubject  toqu,c≤zu,c,z∈Rm×R+×R.One can notice that problemsP4.2 and ED∗ have the same structure. Therefore, we can apply to ED∗ the ideas used by NFDA.However, NFDA can only minimize nonsmooth convex functions which have bounded level sets. The convex functionq, in general, does not enjoy this property. Therefore, the IED method uses directions which are suited to the function q we have. Indeed, the IED method takes advantage of both NFDA search direction and the special features of the dual function q.At the stepk of the IED method, we have a point (vk,zk)=((uk,ck),zk)∈int⁡epiq and a primal iterate xk∈X(vk) such that g(xk)≠0.Now, havingxk, according to Theorem 1, we can easily find a subgradient (21)sk=-Agxk,-σgxk∈∂qvk.This subgradient defines a supporting hyperplane(22)hkv=qvk+sk,v-vkto the epi(q) at the point (vk,q(vk)). By employing this supporting hyperplane, the following auxiliary problem is definedAPkminψu,c,z=zsubject  togku,c,z≤0u,c,z∈Rm×R+×R,where gk:Rm×R+×R→R is given by gk(v,z)=hk(v)-z.Solving two systems, similar to (14)-(15) and (16)-(17), obtained from APk, the directions d1k and d2k are computed. It is important to highlight that there is no duplication of the computational cost since the coefficient matrices of both systems are the same.The search direction is thus defined bydk=d1k+ρkd2k where the parameter ρk>0 must be chosen so that dk is a descent direction for ψ, i.e., (dk)T∇ψ(vk,zk)<0 and, at the same time, a local descent direction for the dual function q. This requires dk to be a descent direction for g as well, i.e., (dk)T∇g(vk,zk)<0.In other words, IED finds a directiond which, at each iteration, belongs to the interior of the cone (23)Kvk,zk=d∈Rm+1×R∣dT∇gvk,zk<0,dT∇ψvk,zk<0,according to Figure 4.Figure 4
The coneK(v,z).Burachik et al. [1] proved that if ρ satisfies (24)-d1T∇gv,zd2T∇gv,z<ρ<1-ξd1T∇ψv,zd2T∇ψv,zthen d∈K(v,z).From the current iterate(vk,zk)∈int⁡(epi(q)), using a suitable stepsize tk, an auxiliary point (25)u~k,c~k,z~k=v~k,z~k=vk,zk+tkdkis computed.Ifg(x~k)=0 where x~k∈X(v~k) then IED stops.Otherwise, i.e., ifg(x~k)≠0 and q(v~k)<z~k then (vk+1,zk+1)=(v~k,z~k).Ifq(v~k)≥z~k then a DSG step is performed in order to obtain the next iterate (vk+1,zk+1)∈int⁡(epi(q)).A new supporting hyperplane at the point(vk+1,qvk+1) is computed, a new auxiliary problem is defined, and the algorithm goes this way until a primal iterate xk satisfying g(xk)=0 is found.We now describe formally the algorithm.


### 5.1. The IED Algorithm

Step 0. Fix a sequence {αk}⊂R+ and a sequence {Bk} of positive definite matrices. Fix β≥η>0, T0∈(0,1), q^≤q¯, where q¯ is the optimal dual value, a symmetric matrix A, and an augmenting function σ. Choose u0∈Rm, c0>0, λ0>0, and z0>q(v0)=q(u0,c0). Take x0∈X(v0) such that g(x0)≠0. Set k=0.Step k. Let (vk,zk)∈int⁡(epi(q)), λk>0, and xk∈X(vk) such that g(xk)≠0. Compute sk=(-A(g(xk)),-σ(g(xk)).Step k.1. Set i=0. For (vk,zk,λk), find d1k, d2k, λ1k, and λ2k by solving systems (26)Bkd1+∇gxkλ1=-∇ψuk,ck,zkΛk∇gxkTd1+Gxkλ1=0,Bkd2+∇gxkλ2=0Λk∇gxkTd2+Gxkλ2=-Λk.Findρ1k, ρ2k satisfying ρ1k = -(d1k)T∇g(vk,zk)/(d2k)T∇g(vk,zk) < ρk≤(1-ξk)((d1k)T∇ψ(vk,zk)/(d2k)T∇ψ(vk,zk)) < ρ2k, with ξk∈(1/3)(1-ρ1k/ρ2k),(2/3)(1-ρ1k/ρ2k)⊂(0,1), and the search direction dk=d1k+ρkd2k.Compute the stepsizetk=max⁡T0,(zk-q^)/-∇ψ(vk,zk)Tdk,|(vk)Tsk|/-∇ψ(vk,zk)Tdk.Step k.2. Compute the auxiliary point (v^k,z^k)=(vk,zk)+tkdk. Find x^k∈X(v^k,z^k). If g(x^k)=0, stop and set xk=x^k and vk=v^k.Step k.3.a (serious step). If the auxiliary point (v^k,z^k)∈int⁡epiq then set (vk+1,zk+1)=(v^k,z^k), xk+1=x~k, λk+1=λk+tk(λ1k+ρkλ2k)=λ^k, k=k+1, and go to Step k.Step k.3.b (null step). Do i=i+1. Choose rk∈[ηk,βk] where ηk=min⁡η,σgx and βk=max⁡β,σgx. Compute a new point in int⁡(epi(q)) by performing a DSG step from vk:(27)u~k,i=uk-rkAgxk;c~k,i=ck-1+αkrkσgxk;v~k,i=u~k,i,c~k,i.Findx~k,i∈X(v~k,i). If g(x~k,i)=0, stop. Otherwise, compute q(v~k,i) and take z~k,i such that q(v~k,i)<z~k,i<q(vk). Set (vk,zk)=(v~k,i,z~k,i); xk=x~k,i, and go to Step k.Figure5 shows the serious step (a) and the null step (b) of the IED algorithm.Figure 5
The IED Algorithm.
(a)
Serious step:vk+1,zk+1=v^k,z^k (b)
Null step:vk,zk=v~k,i,z~k,iAs proved in [1], for the convergence of the method, the matrix Bk∈R(m+2)×(m+2), k∈N, must have the following structure: (28)Bk=Ck00ak,where Ck∈R(m+1)×(m+1) is positive definite. Also, there must exist positive numbers σ1 and σ2 such that (29)σ1w2≤wTCkw≤σ2w2,∀w∈Rm+1.Moreover, there must exist a¯>0 so that ak>a¯ for all k∈N.To finalize this section, we present the result that justifies the stopping criterion used by the method.Theorem 2 (see [2], Theorem 2.2).
Suppose that for some(u¯,c¯)∈Rm×R+ we have x¯∈X(u¯,c¯). Then x¯ is a solution of P and (u¯,c¯) is a solution of D if and only if g(x¯)=0.Theorem2 shows that a minimizer of the Lagrangian not only is the solution of P but is also used to stop the algorithm.


## 5.1. The IED Algorithm

Step 0. Fix a sequence {αk}⊂R+ and a sequence {Bk} of positive definite matrices. Fix β≥η>0, T0∈(0,1), q^≤q¯, where q¯ is the optimal dual value, a symmetric matrix A, and an augmenting function σ. Choose u0∈Rm, c0>0, λ0>0, and z0>q(v0)=q(u0,c0). Take x0∈X(v0) such that g(x0)≠0. Set k=0.Step k. Let (vk,zk)∈int⁡(epi(q)), λk>0, and xk∈X(vk) such that g(xk)≠0. Compute sk=(-A(g(xk)),-σ(g(xk)).Step k.1. Set i=0. For (vk,zk,λk), find d1k, d2k, λ1k, and λ2k by solving systems (26)Bkd1+∇gxkλ1=-∇ψuk,ck,zkΛk∇gxkTd1+Gxkλ1=0,Bkd2+∇gxkλ2=0Λk∇gxkTd2+Gxkλ2=-Λk.Findρ1k, ρ2k satisfying ρ1k = -(d1k)T∇g(vk,zk)/(d2k)T∇g(vk,zk) < ρk≤(1-ξk)((d1k)T∇ψ(vk,zk)/(d2k)T∇ψ(vk,zk)) < ρ2k, with ξk∈(1/3)(1-ρ1k/ρ2k),(2/3)(1-ρ1k/ρ2k)⊂(0,1), and the search direction dk=d1k+ρkd2k.Compute the stepsizetk=max⁡T0,(zk-q^)/-∇ψ(vk,zk)Tdk,|(vk)Tsk|/-∇ψ(vk,zk)Tdk.Step k.2. Compute the auxiliary point (v^k,z^k)=(vk,zk)+tkdk. Find x^k∈X(v^k,z^k). If g(x^k)=0, stop and set xk=x^k and vk=v^k.Step k.3.a (serious step). If the auxiliary point (v^k,z^k)∈int⁡epiq then set (vk+1,zk+1)=(v^k,z^k), xk+1=x~k, λk+1=λk+tk(λ1k+ρkλ2k)=λ^k, k=k+1, and go to Step k.Step k.3.b (null step). Do i=i+1. Choose rk∈[ηk,βk] where ηk=min⁡η,σgx and βk=max⁡β,σgx. Compute a new point in int⁡(epi(q)) by performing a DSG step from vk:(27)u~k,i=uk-rkAgxk;c~k,i=ck-1+αkrkσgxk;v~k,i=u~k,i,c~k,i.Findx~k,i∈X(v~k,i). If g(x~k,i)=0, stop. Otherwise, compute q(v~k,i) and take z~k,i such that q(v~k,i)<z~k,i<q(vk). Set (vk,zk)=(v~k,i,z~k,i); xk=x~k,i, and go to Step k.Figure5 shows the serious step (a) and the null step (b) of the IED algorithm.Figure 5
The IED Algorithm.
(a)
Serious step:vk+1,zk+1=v^k,z^k (b)
Null step:vk,zk=v~k,i,z~k,iAs proved in [1], for the convergence of the method, the matrix Bk∈R(m+2)×(m+2), k∈N, must have the following structure: (28)Bk=Ck00ak,where Ck∈R(m+1)×(m+1) is positive definite. Also, there must exist positive numbers σ1 and σ2 such that (29)σ1w2≤wTCkw≤σ2w2,∀w∈Rm+1.Moreover, there must exist a¯>0 so that ak>a¯ for all k∈N.To finalize this section, we present the result that justifies the stopping criterion used by the method.Theorem 2 (see [2], Theorem 2.2).
Suppose that for some(u¯,c¯)∈Rm×R+ we have x¯∈X(u¯,c¯). Then x¯ is a solution of P and (u¯,c¯) is a solution of D if and only if g(x¯)=0.Theorem2 shows that a minimizer of the Lagrangian not only is the solution of P but is also used to stop the algorithm.


## 6. The Genetic Algorithm

Genetic algorithms (GAs) [20–22] were inspired by the Darwinian principles of natural selection and evolution. They are considered the most famous bioinspired metaheuristic and the basis of innumerous algorithms found in the related literature. GAs encode all the variables, using an adopted alphabet, as binary code, for example, corresponding to a candidate solution (defined as the phenotype) in a data structure (defined as the genotype). A population of genotypes evolves mimicking the evolutionary process found in the Nature. The candidate solutions are submitted to the evaluation of their quality and they are selected by a stochastic process favoring better solutions. The genetic materials of these are recombined and mutated by means of genetic operators generating a new population.Basically, a GA presents the following five steps [21]: (i) a genetic coding of solutions to the problem; (ii) a procedure for creating an initial population of solutions; (iii) an evaluation function that returns the quality of each individual (fitness function); (iv) genetic operators manipulating genetic material of parents during the reproduction process giving rise to new individuals; and (v) parameters to be used in the algorithm during the recombination and mutation operations.The GAs are free-derivative algorithms and massively used to solve unconstrained and constrained optimization problems.The link between IED and Genetic Algorithm recognized through the notationxk∈X(vK) can be seen in Figures 6 and 7.Figure 6
A flow chart for a step of the IED method with a Genetic Algorithms.Figure 7
A flow chart of the Genetic Algorithm used inside the IED.Figure 8
The Tension/Compression String.The pseudocode depicted in Algorithm1 describes the main steps of a generational Genetic Algorithm for minimizing the Lagrangian.Algorithm 1:The generational Genetic Algorithm to minimize the Lagrangian function.
(1) Input:  Parameter popsize, gensize, u, c; (2) Output:   The best solution found in all generations; (3) gen=1; (4) Pop[gen] = InitializePopulation(); (5) Evaluate (L(x,(u,c))=f(x)+cσ(g(x))+〈Au,g(x)〉, Pop[gen]); (6) while  gen≤gensize  do (7) selPop[gen] = Selection(Pop[gen]); (8) newPop[gen] = Crossover(selPop[gen]); (9) Mutation(newPop[gen]); (10) Evaluate (L(x,(u,c))=f(x)+cσ(g(x))+〈Au,g(x)〉, newPop[gen]); (11) Pop[gen+1] = NewGeneration(newPop[gen], Pop[gen]); (12) gen=gen+1; (13) endIn this paper, a Genetic Algorithm with a real code is adopted and the Simulated Binary Crossover (SBX) [23] and the polynomial mutation [24] have been set as the genetic operators. The next subsections describe these operators. The baseline real-coded generational GA uses a rank-based selection and elitism (the best element is always copied into the next generation avoiding the loss of good solutions).


### 6.1. SBX Crossover

SBX is a crossover based on the properties presented by the one-point crossover in binary-coded. These properties are as follows:(i)
Average: the average of the parents is equal to the average of the children after the crossover operation;(ii)
Spread factorβ: most of the crossover events result in a spread factor of β≈1, that is, children tend to be close to their parents.To maintain the average property, the offspring values are calculated as(30)ch1=q¯-12βp2-p1ch2=q¯+12βp2-p1,where q¯=(1/2)(p1+p2), p2>p1, p1 and p2 are the parents and, ch1 and ch2 are the offspring. It can be observed that ch¯=q¯. The parameter β is a random variable obtained from the following probabilistic distribution function:(31)fβ=0.5η+1βη,ifβ≤1,0.5η+11βη+2otherwise,where small values for η generate offspring which are more similar to their parents and, on the other hand, great values generate offspring which are quite different from their parents. In this work, we used η=2, a value that works well according to the original reference [23].


### 6.2. Polynomial Mutation

Polynomial mutation is an operator that uses polynomial probability distribution to perturb a solution in a parent’s proximity. Given a parent solutionp∈[a,b], the polynomial mutation solution p′ is created by using a random number r∈[0,1], for a given variable, as follows:(32)pi′=pi+δ¯Lpi-xiL,forr≤0.5,pi+δ¯RxiR-piforr>0.5,where δ¯L and δ¯R can be calculated as follows: (33)δ¯L=2r1/1+ηm-1,forr≤0.5δ¯R=1-21-r1/1+ηm,forr>0.5.It is important to note that no value outside the range[a,b] is created by the polynomial mutation. In this work, we used ηm=100, a value that works well according to the original reference [24].


## 6.1. SBX Crossover

SBX is a crossover based on the properties presented by the one-point crossover in binary-coded. These properties are as follows:(i)
Average: the average of the parents is equal to the average of the children after the crossover operation;(ii)
Spread factorβ: most of the crossover events result in a spread factor of β≈1, that is, children tend to be close to their parents.To maintain the average property, the offspring values are calculated as(30)ch1=q¯-12βp2-p1ch2=q¯+12βp2-p1,where q¯=(1/2)(p1+p2), p2>p1, p1 and p2 are the parents and, ch1 and ch2 are the offspring. It can be observed that ch¯=q¯. The parameter β is a random variable obtained from the following probabilistic distribution function:(31)fβ=0.5η+1βη,ifβ≤1,0.5η+11βη+2otherwise,where small values for η generate offspring which are more similar to their parents and, on the other hand, great values generate offspring which are quite different from their parents. In this work, we used η=2, a value that works well according to the original reference [23].


## 6.2. Polynomial Mutation

Polynomial mutation is an operator that uses polynomial probability distribution to perturb a solution in a parent’s proximity. Given a parent solutionp∈[a,b], the polynomial mutation solution p′ is created by using a random number r∈[0,1], for a given variable, as follows:(32)pi′=pi+δ¯Lpi-xiL,forr≤0.5,pi+δ¯RxiR-piforr>0.5,where δ¯L and δ¯R can be calculated as follows: (33)δ¯L=2r1/1+ηm-1,forr≤0.5δ¯R=1-21-r1/1+ηm,forr>0.5.It is important to note that no value outside the range[a,b] is created by the polynomial mutation. In this work, we used ηm=100, a value that works well according to the original reference [24].


## 7. Numerical Experiments

In this section, we present the results obtained by the current version of the IED method which uses genetic algorithms for minimizing the Lagrangian.Regarding the functionσ (see (1)), we have used seven norms in our first set of numerical experiments, namely, ·1, ·2, ·∞, e·2-1, e·1-1, me·∞-1, and max(·2,·22). Usually, those norms are used in order to check the efficiency of algorithm [1, 2, 25]. Therefore, we have kept the norms to compare the results obtained by our algorithm.With respect to the matrixA, we have considered A=I which corresponds to the augmented Lagrangian case.Inequality constraintsg(x)≤0 have been replaced with their nonsmooth equivalent max⁡gx,0=0 so that the problems have been converted into equality constrained optimization problems.As the stopping criterion for IED we have usedg(x)≤10-6. A total of 10 independent runs have been executed by the IED + GA, extracting the best solution, the worst solution, the mean of the solutions, and the standard deviation (sd), for the first set of experiments. For the second set, we have executed 30 independent runs.The algorithms have been implemented in Matlab -(R2012b) - in a microcomputer core (TM) i7 of 2.60 GHz with 8.00 GB of RAM.This section is divided into two sets of test problems.


### 7.1. The First Set of Numerical Experiments

This set of numerical experiments corresponds to the analysis of 20 constrained optimization problems extracted from the Hock and Schittkowski collection [26] and all of them refer to minimization problems. They are described below were the objective f(x) and the constraints are presented.GLR-P1-1:(34)fx=-32,174255ln⁡x1+x2+x3+0,030,09x1+x2+x3+0,03+280ln⁡x2+x3+0,030,07x2+x3+0,03+290ln⁡x3+0,030,13x3+0,03subject to(35)x1+x2+x3=00≤xi≤1,i=1,2,3GQR-P1-1: (36)fx=-75,196+3,8112x1+0,0020567x13-1,0345e-5x14+6,8306x2-0,030234x1x2+1,28134e-3x2x12+2,266e-7x14x2-0,25645x22+0,0034604x23-1,3514e-5x24+28,106x2+1+5,2375e-6x12x22+6,3e-8x13x22-7e-10x13x23-3,405e-4x1x22+1,6638e-6x1x23+2,8673exp⁡0,0005x1x2-3,5256e-5x13x2subject to(37)x1x2-700≥0x2-x12125≥0x2-502-5x1-55≥00≤x1≤750≤x2≤65PPR-P1-2: (38)fx=5x1+50000x1+20x2+72000x2+10x3+144000x3subject to(39)1-4x1-32x2-120x31·10-5≤xi,i=1,2,3PQR-T1-7: (40)fx=-x1x2x3subject to(41)-x12-2x22-4x32+48≥0SQR-P1-1: (42)fx=∑i=144fix2,fix=bi-x1-0,49-x1exp⁡-x2ai-8,i=1,…,44subject to(43)0,49x2-x1x2-0,09≥00,4≤x1-4≤x2LGR-P1-1:(44)fx=0,2x3-0,8x1subject to(45)x2-exp⁡x1≥0x3-exp⁡x2≥00≤x1≤1000≤x2≤1000≤x3≤10QBR-T1-1: (46)fx=x2+10-5x2-x12subject to(47)0≤x2PBR-T1-1: (48)fx=100x2-x122+1-x12subject to(49)-x2-1,5≤0PBR-T1-2: (50)fx=100x2-x122+1-x12subject to(51)1,5≤x2QQR-P1-3: (52)fx=x1-x22+x1+x2-1029+x3-52subject to(53)48-x12-x22-x32≥0-4,5≤xi≤4,5,i=1,2-5≤x3≤5QQR-T1-6: (54)fx=x1-22+x2-12subject to(55)x12-x2≤0x1+x2-2≤0PLR-T1-1: (56)fx=1273x1-32-9x23subject to(57)x13-x2≥0x1+3x2≥0-x1+3x2-6≥00≤x10≤x2PBR-T1-3: (58)fx=13x1+13+x2subject to(59)1≤x10≤x2QLR-T1-1: (60)fx=0,01x12+x22-100subject to(61)10x1-x2-10≥02≤x1≤50-50≤x2≤50PQR-T1-1: (62)fx=100x2-x122+1-x12subject to(63)x1x2-1≥0x1+x22≥0x1≤0,5QQR-T1-3: (64)fx=0,5x12+x22-x1x2-7x1-7x2subject to(65)-25+4x12+x22≤0GBR-T1-1: (66)fx=sin⁡x1+x2+x1-x22-1,5x1+2,5x2+1subject to(67)-1,5≤x1≤4-3≤x2≤3PQR-T1-4: (68)fx=x1-103+x2-203subject to(69)-100+x1-52+x2-52≥0-x2-52-x1-62+82.81≥013≤x1≤1000≤x2≤100QQR-T1-2: (70)fx=x1-52+x22-25subject to(71)-x12+x22≥0QPR-T1-1: (72)fx=x1-22+x22subject to(73)1-x13-x2≥00≤x10≤x2


#### 7.1.1. Comparison of Results of the First Set of Numerical Experiments

Table1 provides the initial guesses c0 and u0 used by IED + fmin and IED + GA, the size of the population, the number of generations, and bound constraints used by the Genetic Algorithm. The table also shows the optimal values.Table 1
Parameters of the IED method.
Test-problem c 0 u 0 Population Size Generations Lower Bound Upper Bound Optimal Value GLR-P1-1 50 [ 0   0   0   0 ] 50 300 [ 0.01 ,   0.01 ,   0.01 ] [ 1 ,   1 ,   1 ] -2.6273×104 GQR-P1-1 1 [ 1   1   1   1   1   1   1 ] 50 100 [ 10 ,   10 ] [ 60 ,   60 ] -7.8028 PPR-P1-2 0.1 [ 1   1   1   1 ] 50 200 [ 0 ,   0 ,   0 ] [ 300 ,   300 ,   300 ] 6.2998×103 PQR-T1-7 1.5 [ 1   1   1   1   1   1   1 ] 50 100 [ - 5 ,   - 5 ,   - 5 ] [ 5 ,   5 ,   5 ] -22.6274 SQR-P1-1 0.2 [ 0   0   0 ] 50 100 [ - 10 ,   - 10 ] [ 10 ,   10 ] 0.0285 LGR-P1-1 0.1 [ 0   0   0   0   0   0   0   0 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0.5181 QBR-T1-1 1 [ 1   1   1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0 PBR-T1-1 1 [ 1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 0 PBR-T1-2 1 [ 1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 0.0504 QQR-P1-3 1 [ 1   1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0.9535 QQR-T1-6 1 [ 1   1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 1 PLR-T1-1 1 [ 1   1   1   1   1 ] 50 100 [ 1 ,   1 ] [ 4 ,   4 ] −1 PBR-T1-3 1 [ 1   1 ] 50 100 [ - 1 ,   - 1 ] [ 2 ,   2 ] 2.6666 QLR-T1-1 1 [ 1   1   1   1   1 ] 50 100 [ - 1 ,   - 1 ] [ 3 ,   3 ] −99.96 PQR-T1-1 5 [ 1   1   1 ] 50 100 [ 0 ,   0 ] [ 3 ,   3 ] 306.5 QQR-T1-3 0.1 [ 1 ] 20 50 [ 1 ,   1 ] [ 4 ,   4 ] −30 GBR-T1-1 1 [ 1   1   1   1 ] 50 100 [ - 2 ,   - 2 ] [ 0 ,   0 ] −1.9132 PQR-T1-4 1 [ 1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ] [ 15 ,   15 ] −6961.8138 QQR-T1-2 1 [ 1 ] 20 50 [ 0 ,   0 ] [ 2 ,   2 ] −8.4984 QPR-T1-1 0.1 [ 1   1   1 ] 50 100 [ - 1 ,   - 1 ] [ 2 ,   2 ] 1For solving the first six problems, we have kept the initial guesses used by the IED in its previous version as described in [1]. For the rest of the problems, c0 and u0 were randomly chosen.One can notice from Tables2–21 that IED + GA has found the solution for all the problems regardless of the norms used whereas IED + fmin failed some of them.Table 2
Comparison of the results obtained by both versions of the IED for problem GLR-P1-1.
σ ( ⋅ ) GLR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −2.6273 ×104 - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 ⋅ 2 - 2.6273 × 1 0 4 - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 m ⋅ ∞ — - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 e ⋅ 2 - 1 - 2.6273 × 1 0 4 - 2.6273 × 1 0 4 - 2.6270 × 1 0 4 - 2.6272 × 1 0 4 0.8555 e ⋅ 1 - 1 - 2.6273 × 1 0 4 - 2.6273 × 1 0 4 - 2.6270 × 1 0 4 - 2.6272 × 1 0 4 0.8555 m(e⋅∞-1) — - 2.6272 × 1 0 4 - 2.6225 × 1 0 4 - 2.6261 × 1 0 4 14.5763 m a x ( ⋅ 2 , ⋅ 2 2 ) — - 2.6273 × 1 0 4 - 2.4727 × 1 0 4 - 2.6049 × 1 0 4 477.8463Table 3
Comparison of the results obtained by both versions of the IED for problem GQR-P1-1.
σ ( ⋅ ) GQR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −7.7978 −6.7495 −7.5575 0.4268 ⋅ 2 — −7.7978 −6.7495 −7.5572 0.4276 m ⋅ ∞ −7.8028 −7.7984 −6.7495 −7.0554 0.4927 e ⋅ 2 - 1 −7.8028 −7.7925 −6.7495 −7.4544 0.4872 e ⋅ 1 - 1 −7.8028 −7.8012 −7.7443 −7.7801 0.0171 m(e⋅∞-1) −7.8028 −7.7985 −6.7495 −7.4601 0.4914 m a x ( ⋅ 2 , ⋅ 2 2 ) −7.8028 −7.8024 −6.7495 −7.3560 0.5228Table 4
Comparison of the results obtained by both versions of the IED for problem PPR-P1-2.
σ ( ⋅ ) PPR-P1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 6.2998 × 1 0 3 6.3002 × 1 0 3 6.3018 × 1 0 3 6.3008 × 1 0 3 0.4452 ⋅ 2 6.2998 × 1 0 3 6.3002 × 1 0 3 6.3018 × 1 0 3 6.3008 × 1 0 3 0.4452 m ⋅ ∞ 6.2998 × 1 0 3 6.3001 × 1 0 3 6.3035 × 1 0 3 6.3015 × 1 0 3 1.0803 e ⋅ 2 - 1 — 6.3000 × 1 0 3 6.3023 × 1 0 3 6.3007 × 1 0 3 0.7107 e ⋅ 1 - 1 — 6.3000 × 1 0 3 6.3023 × 1 0 3 6.3007 × 1 0 3 0.7107 m(e⋅∞-1) — 6.3000 × 1 0 3 6.3053 × 1 0 3 6.3012 × 1 0 3 1.6442 m a x ( ⋅ 2 , ⋅ 2 2 ) 6.2999 × 1 0 3 6.2999 × 1 0 3 6.3045 × 1 0 3 6.3013 × 1 0 3 1.5406Table 5
Comparison of the results obtained by both versions of the IED for problem PQR-T1-7.
σ ( ⋅ ) PQR-T1-7 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −22.5467 −22.1224 −22.4160 0.1450 ⋅ 2 — −22.6033 −22.1466 −22.4274 0.1383 m ⋅ ∞ −22.6274 −22.6233 −22.3691 −22.5153 0.1151 e ⋅ 2 - 1 −22.6262 −22.5424 −22.0668 −22.3474 0.1526 e ⋅ 1 - 1 −22.6270 −22.5468 −21.9238 −22.2471 0.1891 m(e⋅∞-1) −22.6274 −22.5215 21.6514 −17.8111 13.8676 m a x ( ⋅ 2 , ⋅ 2 2 ) −22.6156 −22.5704 −22.0331 −22.3381 0.1683Table 6
Comparison of the results obtained by both versions of the IED for problem SQR-P1-1.
σ ( ⋅ ) SQR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.0285 0.0285 0.0287 0.0286 8.4993 × 1 0 - 5 ⋅ 2 0.0285 0.0285 0.0287 0.0286 8.2863 × 1 0 - 5 m ⋅ ∞ 0.0285 0.0285 0.0293 0.0287 2.3577 × 1 0 - 4 e ⋅ 2 - 1 0.0285 0.0285 0.0293 0.0287 5.7703 × 1 0 - 4 e ⋅ 1 - 1 0.0285 0.0285 0.0306 0.0289 6.5958 × 1 0 - 4 m(e⋅∞-1) 0.0285 0.0285 0.0290 0.0286 1.8364 × 1 0 - 4 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.0285 0.0285 0.0289 0.0286 1.2819 × 1 0 - 4Table 7
Comparison of the results obtained by both versions of the IED for problem LGR-P1-1.
σ ( ⋅ ) LGR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.5187 0.5234 0.5407 0.5314 0.0061 ⋅ 2 — 0.5216 0.5427 0.5301 0.0083 m ⋅ ∞ 0.5228 0.5213 0.5726 0.5364 0.0157 e ⋅ 2 - 1 0.5199 0.5213 0.5726 0.5364 0.0157 e ⋅ 1 - 1 0.5195 0.5232 0.5622 0.5384 0.0125 m(e⋅∞-1) 0.51 83 0.5239 0.5677 0.5390 0.0161 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.5182 0.5201 0.5488 0.5313 0.0090Table 8
Comparison of the results obtained by both versions of the IED for problem QBR-T1-1.
σ ( ⋅ ) QBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 ⋅ 2 — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 m ⋅ ∞ — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 e ⋅ 2 - 1 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 e ⋅ 1 - 1 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 m(e⋅∞-1) 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 m a x ( ⋅ 2 , ⋅ 2 2 ) 2.4563 × 1 0 - 14 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5Table 9
Comparison of the results obtained by both versions of the IED for problem PBR-T1-1.
σ ( ⋅ ) PBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 ⋅ 2 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m ⋅ ∞ - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 e ⋅ 2 - 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 e ⋅ 1 - 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m(e⋅∞-1) - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m a x ( ⋅ 2 , ⋅ 2 2 ) - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052Table 10
Comparison of the results obtained by both versions of the IED for problem PBR-T1-2.
σ ( ⋅ ) PBR-T1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.0504 0.0506 0.0677 0.0536 0.0052 ⋅ 2 0.0504 0.0506 0.0677 0.0536 0.0052 m ⋅ ∞ 0.0504 0.0506 0.0677 0.0536 0.0052 e ⋅ 2 - 1 0.0504 0.0508 0.0839 0.0593 0.0123 e ⋅ 1 - 1 0.0504 0.0508 0.0839 0.0593 0.0123 m(e⋅∞-1) 0.0504 0.0508 0.0839 0.0593 0.0123 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.0504 0.0505 0.0790 0.0545 0.0088Table 11
Comparison of the results obtained by both versions of the IED for problem QQR-P1-3.
σ ( ⋅ ) QQR-P1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.9535 0.9576 0.9837 0.9676 0.0099 ⋅ 2 0.9535 0.9565 0.9853 0.9637 0.0890 m ⋅ ∞ 0.9535 0.9550 0.9881 0.9665 0.0115 e ⋅ 2 - 1 0.9535 0.9550 0.9684 0.9603 0.0039 e ⋅ 1 - 1 0.9535 0.9541 0.9860 0.9652 0.0096 m(e⋅∞-1) −25.0694 0.9558 0.9683 0.9628 0.0040 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.9535 0.9568 1.0124 0.9708 0.0185Table 12
Comparison of the results obtained by both versions of the IED for problem QQR-T1-6.
σ ( ⋅ ) QQR-T1-6 IED + fmin IED + GA result best worst mean sd ⋅ 1 1 1.0002 1.0185 1.0052 0.0057 ⋅ 2 1 1.0023 1.0179 1.0101 0.0044 m ⋅ ∞ 1 1.0003 1.0404 1.0112 0.0125 e ⋅ 2 - 1 1 1.0005 1.0287 1.0126 0.0093 e ⋅ 1 - 1 1 1.0005 1.0400 1.0122 0.0137 m(e⋅∞-1) 1 1.0002 1.0366 1.0133 0.0106 m a x ( ⋅ 2 , ⋅ 2 2 ) 1 1.0002 1.0097 1.0046 0.0037Table 13
Comparison of the results obtained by both versions of the IED for problem PLR-T1-1.
σ ( ⋅ ) PLR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −0.9976 −0.9729 −0.9915 0.0071 ⋅ 2 — −0.9966 −0.9817 −0.9892 0.0052 m ⋅ ∞ — −0.9984 −0.9663 −0.9899 0.0102 e ⋅ 2 - 1 −1 −0.9974 −0.9806 −0.9916 0.0057 e ⋅ 1 - 1 −1 −0.9966 −0.9805 −0.9925 0.0048 m(e⋅∞-1) −1 −0.9993 −0.9793 −0.9926 0.0061 m a x ( ⋅ 2 , ⋅ 2 2 ) −1 −0.9996 −0.9758 −0.9920 0.0073Table 14
Comparison of the results obtained by both versions of the IED for problem PBR-T1-3.
σ ( ⋅ ) PBR-T1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 2.6686 2.7135 2.6827 0.0134 ⋅ 2 — 2.6698 2.6855 2.6743 0.0050 m ⋅ ∞ — 2.6697 2.6909 2.6784 0.0079 e ⋅ 2 - 1 2.6667 2.6692 2.6947 2.6784 0.0082 e ⋅ 1 - 1 2.6667 2.6685 2.7022 2.6809 0.0110 m(e⋅∞-1) 2.6667 2.6687 2.7059 2.6909 0.0125 m a x ( ⋅ 2 , ⋅ 2 2 ) 2.6667 2.6684 2.7125 2.6847 0.0150Table 15
Comparison of the results obtained by both versions of the IED for problem QLR-T1-1.
σ ( ⋅ ) QLR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −99.9600 −99.9600 −99.9597 −99.9599 8.9350 × 1 0 - 5 ⋅ 2 −99.9600 −99.9600 −99.9595 −99.9598 1.4311 × 1 0 - 4 m ⋅ ∞ −99.9600 −99.9600 −99.9599 −99.9599 5.1418 × 1 0 - 5 e ⋅ 2 - 1 −99.9600 −99.9600 −99.9594 −99.9599 1.5289 × 1 0 - 4 e ⋅ 1 - 1 −99.9600 −99.9600 −99.9595 −99.9599 1.3253 × 1 0 - 4 m(e⋅∞-1) −99.9600 −99.9600 −99.9594 −99.9598 1.8280 × 1 0 - 4 m a x ( ⋅ 2 , ⋅ 2 2 ) −99.9600 −99.9599 −99.9598 −99.9599 3.7864 × 1 0 - 5Table 16
Comparison of the results obtained by both versions of the IED for problem PQR-T1-1.
σ ( ⋅ ) PQR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 306.5 306.5146 326.6933 310.9483 6.2614 ⋅ 2 306.5 308.7375 329.3517 312.3063 6.1269 m ⋅ ∞ 306.5 307.5067 327.2424 311.6510 5.8228 e ⋅ 2 - 1 306.5 308.9435 317.7634 312.9006 2.7421 e ⋅ 1 - 1 306.5 306.5408 315.6229 310.7872 3.0505 m(e⋅∞-1) 306.5 307.8027 316.7068 312.1473 2.7962 m a x ( ⋅ 2 , ⋅ 2 2 ) 306.5 307.9935 315.4329 311.4265 2.4128Table 17
Comparison of the results obtained by both versions of the IED for problem QQR-T1-3.
σ ( ⋅ ) QQR-T1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −29.9998 −29.8286 −29.9714 0.0532 ⋅ 2 — −29.9998 −29.8286 −29.9714 0.532 m ⋅ ∞ — −29.9998 −29.8286 −29.9714 0.0532 e ⋅ 2 - 1 −30 −29.9972 −29.3533 −29.8566 0.2162 e ⋅ 1 - 1 −30 −29.9972 −29.3533 −29.8566 0.2162 m(e⋅∞-1) −30 −29.9972 −29.3533 −29.8566 0.2162 m a x ( ⋅ 2 , ⋅ 2 2 ) −30 −29.9960 −29.2340 −29.8780 0.2316Table 18
Comparison of the results obtained by both versions of the IED for problem GBR-T1-1.
σ ( ⋅ ) GBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7 ⋅ 2 −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7 m ⋅ ∞ −1.9132 −1.9132 −1.9132 −1.9132 1.4981 × 1 0 - 6 e ⋅ 2 - 1 −1.9132 −1.9132 −1.9132 −1.9132 9.3127 × 1 0 - 7 e ⋅ 1 - 1 −1.9132 −1.9132 −1.9132 −1.9132 9.3127 × 1 0 - 7 m(e⋅∞-1) −1.9132 −1.9132 −1.9132 −1.9132 1.2166 × 1 0 - 6 m a x ( ⋅ 2 , ⋅ 2 2 ) −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7Table 19
Comparison of the results obtained by both versions of the IED for problem PQR-T1-4.
σ ( ⋅ ) PQR-T1-4 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −6791.5 173850 1237.6 5.6747 × 1 0 4 ⋅ 2 — −6936.9 1792.1 −5153.5 2.8982 × 1 0 3 m ⋅ ∞ — −6800.2 −2548.9 −5962.7 1.3694 × 1 0 3 e ⋅ 2 - 1 −6961.8 −6874.3 −6258.0 −6737.2 188.5621 e ⋅ 1 - 1 −6961.8 −6915.8 −1642.2 −6165.1 1606.1 m(e⋅∞-1) −6961.8 −6777.9 −6159.4 −6590.4 222.8402 m a x ( ⋅ 2 , ⋅ 2 2 ) −6961.8 −6958.0 −1735.3 −6274.8 1609.0Table 20
Comparison of the results obtained by both versions of the IED for problem QQR-T1-2.
σ ( ⋅ ) QQR-T1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 −8.4985 −8.4958 −8.3391 −8.4729 0.0481 ⋅ 2 −8.4985 −8.4958 −8.3391 −8.4729 0.0481 m ⋅ ∞ −8.4985 −8.4958 −8.3391 −8.4729 0.0481 e ⋅ 2 - 1 −8.4985 −8.4958 −7.5694 −8.2675 0.3201 e ⋅ 1 - 1 −8.4985 −8.4958 −7.5694 −8.2675 0.3201 m(e⋅∞-1) −8.4985 −8.4958 −7.5694 −8.2675 0.3201 m a x ( ⋅ 2 , ⋅ 2 2 ) −8.4985 −8.4980 −8.0434 −8.3737 0.1564Table 21
Comparison of the results obtained by both versions of the IED for problem QPR-T1-1.
σ ( ⋅ ) QPR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 1.1349 1.4578 1.2706 0.0846 ⋅ 2 — 1.0632 1.4943 1.2132 0.1356 m ⋅ ∞ — 1.1317 1.6091 1.2670 0.1421 e ⋅ 2 - 1 0.9978 1.1316 1.4839 1.2715 0.1220 e ⋅ 1 - 1 — 1.0729 1.5440 1.2488 0.1375 m(e⋅∞-1) 0.9979 1.0818 1.4799 1.2300 0.1555 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.9987 1.0967 1.6843 1.2762 0.1853It is interesting to highlight that the standard deviation of all the problems is small. This shows robustness of the version of the IED method proposed in this work.Tables2–21 show the results of the problems obtained by both versions of the algorithm. For the IED + GA algorithm, we present the best and the worst results, the mean and the standard deviation. Cases where a solution has been not found are indicated by a line struck through the entry of the table.


### 7.2. The Second Set of Numerical Experiments

In this section the algorithm proposed is applied to solve 4 real-world mechanical engineering optimization problems, largely used as a test-bed in the literature.The Tension/Compression String Design. In this problem, the objective is to minimize the volume V of a coil spring, depicted in the Figure 8, under a constant tension/compression load. There are three design variables to be considered in this problem: The number x1=N of active coils of the spring, the winding diameter x2=D and the wire diameter x3=d. The volume of the coil to be minimized is written as [27] (74)Vx=x1+2x2x32and is subject to the constraints (75)g1x=1-x23x171785x34≤0g2x=4x22-x3x212566x2x33-x34+15108x32-1≤0g3x=1-140.45x3x22x1≤0g4x=x2+x31.5-1≤0,where (76)2≤x1≤150.25≤x2≤1.30.05≤x3≤2.The Speed Reducer Design. The objective of this problem is to minimize the weight W of the speed reducer [27] shown in the Figure 9. The design variables are the face width (x1=b), the module of teeth (x2=m), the number of teeth on pinion (x3=n), the length of the shaft 1 between the bearings (x4=l1), the length of the shaft 2 between the bearings (x5=l2), the diameter of the shaft 1 (x6=d1), and, finally, the diameter of the shaft 2 (x7=d2). The third variable is integer and all the others are continuous. The constraints include limitations on the bending and surface stress of the gear teeth, transverse deflections of the shafts 1 and 2 generated by the transmitted force, and, finally, the stress in the shafts 1 and 2. The weight of the speed reducer, to be minimized, is given by(77)Wx=0.7854x1x223.3333x32+14.9334x3-43.0934-1.508x1x62+x72+7.477x63+x73+0.7854x4x62+x5x72subject to (78)g1x=27x1-1x2-2x3-1≤1g2x=397.5x1-1x2-2x3-2≤1g3x=1.93x2-1x3-1x43x6-4≤1g4x=1.93x2-1x3-1x53x7-4≤1g5x=745x4/x2x32+16.9×1060.50.1x63≤1100g6x=745x5/x2x32+157.5×1060.50.1x73≤850g7x=x2x3≤40g8x=x1x2≥5g9x=x1x2≤12g10x=1.5x6+1.9x4-1≤1g11x=1.1x7+1.9x5-1≤12.6≤x1≤3.60.7≤x2≤0.817≤x3≤287.3≤x4≤8.37.8≤x5≤8.32.9≤x6≤3.95.0≤x6≤5.5.Figure 9
The Speed Reducer design.The Welded Beam. This problem corresponds to the design of the welded beam [27] depicted in the Figure 10. The design variables are {h,l,t,b}, with bounds 0.125≤h≤10, and 0.1≤l,t,b≤10. The objective function to be minimized is the cost of the beam given as (79)Ch,l,t,b=1.10471h2l+0.04811tb14.0+lsubject to (80)g1τ=1-τ13600≥0g2σ=1-σ30000≥0g3b,h=b-h≥0g4Pc=Pc6000-1≥0g5δ=0.25-δ≥0.The expressions for τ, σ, Pc, and δ involve the design variables and are given by (81)τ=τ′2+τ′′2+lτ′τ′′αα=0.25l2+h+t2,σ=504000t2bPc=64746.0221-0.0282346ttb3δ=2.1952t3b,τ′=60002hlτ′′=600014+0.5lα20.707hll2/12+0.25h+t2.Figure 10
The welded beam.The Pressure Vessel. This problem corresponds to the weight minimization of a cylindrical pressure vessel with two spherical heads [27] as shown in Figure 11. The objective function involves four variables: the thickness of the pressure vessel (Ts), the thickness of the head (Th), the inner radius of the vessel (R), and the length of the cylindrical component (L). Since there are two discrete variables (Ts and Th) and two continuous variables (R and L), one has a nonlinearly constrained mixed discrete-continuous optimization problem.Figure 11
The pressure vessel.The bounds of the design variables are0,0625≤Ts,Th≤5 (in constant steps of 0.0625) and 10≤R,L≤200. The design variables are given in inches and the weight is written as (82)wTs,Th,R,L=0,6224TsThRL+1.7781ThR2+3.1661Ts2L+19.84Ts2Rto be minimized subject to the constraints (83)g1Ts,R=Ts-0.0193R≥0g2Th,R=Th-0.00954R≥0g3R,L=πR2L+4/3πR31296000-1≥0g4L=-L+240≥0.The first two constraints establish a lower bound to the ratios Ts/R and Th/R, respectively. The third constraint corresponds to a lower bound for the volume of the vessel and the last one to an upper bound to length of the cylindrical component.


#### 7.2.1. Comparison of Results of the Second Set of Numerical Experiments

Table22 provides the initial guesses c0 and u0 used by IED + GA, the size of the population, and the number of generations used by the Genetic Algorithm.Table 22
Parameters of the IED method.
Problems c 0 u 0 Population Size Generations The Tension/Compression String 10 [ 1   1   1   1 ] 100 300 The Speed Reducer 1 [ 1   1   1   1   1   1   1   1   1   1   1 ] 100 300 The Welded Beam 10 [ 1   1   1   1   1 ] 100 300 The Pressure Vessel 1 [ 1   1   1   1 ] 100 300Table23 presents the best and the worst results, the mean, and the standard deviation. Tables 24–27 show the results found by the algorithm.Table 23
Statistical Analysis obtained by IED + GA in the engineering problems.
Engineering problems IED + GA best worst mean sd The Tension/Compression String 0.0128 0.0164 0.0144 0.0009 The Speed Reducer 2996.7 3039.7 3015.0 13.7485 The Welded Beam 2.4151 5.4177 3.1995 0.7005 The Pressure Vessel 6477.2 8787.4 7454.1 636.1933Table 24
Comparison for The Tension/Compression String problem.
Var. Alg-1 [11] Alg-2 [11] IED + GA x 1 11.852177 11.329555 11.184056 x 2 0.34747463 0.35603234 0.36004951 x 3 0.051301897 0.051660806 0.051889349 g 1 - 0.00000012 - 0.000006437 - 0.003088921 g 2 - 0.00000047 - 0.000013709 - 0.003358194 g 3 - 4.03513200 - 4.052324300 - 4.026621039 g 4 - 0.73414900 - 0.728204600 - 0.725374090  V 0.012668 0.012666 0.012781Table 25
Comparison for the speed reducer problem.
Var. Ref. [12] Ref. [13] Alg-1 [11] Alg-2 [11] IED + GA x 1 3.506163 3.500000 3.500001 3.5 3.499684 x 2 0.700831 0.700000 0.700000 0.7 0.700011 x 3 17.0 17.0 17.0 17.0 17.290709 x 4 7.460181 7.300008 7.300017 7.3000035 7.302923 x 5 7.962143 7.715322 7.715326 7.7153225 7.806459 x 6 3.362900 3.350215 3.350216 3.3502147 3.350185 x 7 5.308949 5.286655 5.286654 5.2866545 5.286355 g 1 - 0.077734 - 0.07391524 - 0.07391554 - 0.07391524 - 0.07386154 g 2 - 0.201305 - 0.19799852 - 0.19799876 - 0.19799852 - 0.19795199 g 3 - 0.474119 - 0.49917084 - 0.49916983 - 0.49917156 - 0.49856110 g 4 - 0.897068 - 0.90464383 - 0.90464365 - 0.90464383 - 0.90120377 g 5 - 0.011021 - 0.00000023 - 0.00000155 - 0.000000119 0.000030538 g 6 - 0.012500 - 0.00000029 0.00000000 0.00000000 0.000187373 g 7 - 0.702147 - 0.70250000 - 0.70250000 - 0.702500000 - 0.702495228 g 8 - 0.000573 0.00000000 - 0.00000029 0.00000000 0.00010613 g 9 - 0.583095 - 0.5833333 - 0.58333320 - 0.583333300 - 0.583377553 g 10 - 0.069144 - 0.051326692 - 0.05132753 - 0.051326156 - 0.051711423 g 11 - 0.027920 - 0.00000018 - 0.00000077 - 0.000000357 - 0.011717075  V 3025.0051 2994.4717 2994.4720 2994.4712 2996.7085Table 26
Comparison for the welded beam problem.
Var. Ref. [14] Alg-1 [11] Alg-2 Ref. [11] IED + GA x1 0.2442949 0.24432427 0.24438575 0.240091515 x2 6.2116738 6.2201996 6.2183037 6.390631032 x3 8.3015486 8.291464 8.291165 8.314142911 x4 0.2443003 0.24436942 0.24438748 0.246214266 g1 0.0004447 0.000000000 0.001953125 0.004494538 g2 64.378068 0.001953125 0.056640625 0.012900125 g3 0.0000054 0.000045150 0.000001728 0.006122751 g4 0.0002553 0.029785156 1.210937500 0.024765021 g5 0.2342937 0.234240830 0.234240280 0.234486548  V 2.38159 2.381246 2.3812175 2.4151069Table 27
Comparison for the pressure vessel problem.
Var. Ref. [13] Ref. [14] Alg-1 [11] Alg-2 [11] IED + GA Ts 0.8125 0.8125 0.8125 0.8125 0.90625 Th 0.4375 0.4375 0.4375 0.4375 0.46250 R 42.086994 42.0946558 42.093082 42.094967 46.189759 L 176.779128 176.684062 176.70308 176.67972 138.552311 g1 0.000221 0.000073 0.0001035 0.000007 0.014787 g2 0.035990 0.035917 0.0359320 0.035914 0.021849 g3 3.219817 2.929000 0.1562500 0.0625 0.035064 g4 63.220872 63.315938 63.296920 63.320282 101.447688  V 6061.1229 6060.187934 6060.3677 6060.138 6477.182


## 7.1. The First Set of Numerical Experiments

This set of numerical experiments corresponds to the analysis of 20 constrained optimization problems extracted from the Hock and Schittkowski collection [26] and all of them refer to minimization problems. They are described below were the objective f(x) and the constraints are presented.GLR-P1-1:(34)fx=-32,174255ln⁡x1+x2+x3+0,030,09x1+x2+x3+0,03+280ln⁡x2+x3+0,030,07x2+x3+0,03+290ln⁡x3+0,030,13x3+0,03subject to(35)x1+x2+x3=00≤xi≤1,i=1,2,3GQR-P1-1: (36)fx=-75,196+3,8112x1+0,0020567x13-1,0345e-5x14+6,8306x2-0,030234x1x2+1,28134e-3x2x12+2,266e-7x14x2-0,25645x22+0,0034604x23-1,3514e-5x24+28,106x2+1+5,2375e-6x12x22+6,3e-8x13x22-7e-10x13x23-3,405e-4x1x22+1,6638e-6x1x23+2,8673exp⁡0,0005x1x2-3,5256e-5x13x2subject to(37)x1x2-700≥0x2-x12125≥0x2-502-5x1-55≥00≤x1≤750≤x2≤65PPR-P1-2: (38)fx=5x1+50000x1+20x2+72000x2+10x3+144000x3subject to(39)1-4x1-32x2-120x31·10-5≤xi,i=1,2,3PQR-T1-7: (40)fx=-x1x2x3subject to(41)-x12-2x22-4x32+48≥0SQR-P1-1: (42)fx=∑i=144fix2,fix=bi-x1-0,49-x1exp⁡-x2ai-8,i=1,…,44subject to(43)0,49x2-x1x2-0,09≥00,4≤x1-4≤x2LGR-P1-1:(44)fx=0,2x3-0,8x1subject to(45)x2-exp⁡x1≥0x3-exp⁡x2≥00≤x1≤1000≤x2≤1000≤x3≤10QBR-T1-1: (46)fx=x2+10-5x2-x12subject to(47)0≤x2PBR-T1-1: (48)fx=100x2-x122+1-x12subject to(49)-x2-1,5≤0PBR-T1-2: (50)fx=100x2-x122+1-x12subject to(51)1,5≤x2QQR-P1-3: (52)fx=x1-x22+x1+x2-1029+x3-52subject to(53)48-x12-x22-x32≥0-4,5≤xi≤4,5,i=1,2-5≤x3≤5QQR-T1-6: (54)fx=x1-22+x2-12subject to(55)x12-x2≤0x1+x2-2≤0PLR-T1-1: (56)fx=1273x1-32-9x23subject to(57)x13-x2≥0x1+3x2≥0-x1+3x2-6≥00≤x10≤x2PBR-T1-3: (58)fx=13x1+13+x2subject to(59)1≤x10≤x2QLR-T1-1: (60)fx=0,01x12+x22-100subject to(61)10x1-x2-10≥02≤x1≤50-50≤x2≤50PQR-T1-1: (62)fx=100x2-x122+1-x12subject to(63)x1x2-1≥0x1+x22≥0x1≤0,5QQR-T1-3: (64)fx=0,5x12+x22-x1x2-7x1-7x2subject to(65)-25+4x12+x22≤0GBR-T1-1: (66)fx=sin⁡x1+x2+x1-x22-1,5x1+2,5x2+1subject to(67)-1,5≤x1≤4-3≤x2≤3PQR-T1-4: (68)fx=x1-103+x2-203subject to(69)-100+x1-52+x2-52≥0-x2-52-x1-62+82.81≥013≤x1≤1000≤x2≤100QQR-T1-2: (70)fx=x1-52+x22-25subject to(71)-x12+x22≥0QPR-T1-1: (72)fx=x1-22+x22subject to(73)1-x13-x2≥00≤x10≤x2


### 7.1.1. Comparison of Results of the First Set of Numerical Experiments

Table1 provides the initial guesses c0 and u0 used by IED + fmin and IED + GA, the size of the population, the number of generations, and bound constraints used by the Genetic Algorithm. The table also shows the optimal values.Table 1
Parameters of the IED method.
Test-problem c 0 u 0 Population Size Generations Lower Bound Upper Bound Optimal Value GLR-P1-1 50 [ 0   0   0   0 ] 50 300 [ 0.01 ,   0.01 ,   0.01 ] [ 1 ,   1 ,   1 ] -2.6273×104 GQR-P1-1 1 [ 1   1   1   1   1   1   1 ] 50 100 [ 10 ,   10 ] [ 60 ,   60 ] -7.8028 PPR-P1-2 0.1 [ 1   1   1   1 ] 50 200 [ 0 ,   0 ,   0 ] [ 300 ,   300 ,   300 ] 6.2998×103 PQR-T1-7 1.5 [ 1   1   1   1   1   1   1 ] 50 100 [ - 5 ,   - 5 ,   - 5 ] [ 5 ,   5 ,   5 ] -22.6274 SQR-P1-1 0.2 [ 0   0   0 ] 50 100 [ - 10 ,   - 10 ] [ 10 ,   10 ] 0.0285 LGR-P1-1 0.1 [ 0   0   0   0   0   0   0   0 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0.5181 QBR-T1-1 1 [ 1   1   1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0 PBR-T1-1 1 [ 1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 0 PBR-T1-2 1 [ 1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 0.0504 QQR-P1-3 1 [ 1   1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0.9535 QQR-T1-6 1 [ 1   1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 1 PLR-T1-1 1 [ 1   1   1   1   1 ] 50 100 [ 1 ,   1 ] [ 4 ,   4 ] −1 PBR-T1-3 1 [ 1   1 ] 50 100 [ - 1 ,   - 1 ] [ 2 ,   2 ] 2.6666 QLR-T1-1 1 [ 1   1   1   1   1 ] 50 100 [ - 1 ,   - 1 ] [ 3 ,   3 ] −99.96 PQR-T1-1 5 [ 1   1   1 ] 50 100 [ 0 ,   0 ] [ 3 ,   3 ] 306.5 QQR-T1-3 0.1 [ 1 ] 20 50 [ 1 ,   1 ] [ 4 ,   4 ] −30 GBR-T1-1 1 [ 1   1   1   1 ] 50 100 [ - 2 ,   - 2 ] [ 0 ,   0 ] −1.9132 PQR-T1-4 1 [ 1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ] [ 15 ,   15 ] −6961.8138 QQR-T1-2 1 [ 1 ] 20 50 [ 0 ,   0 ] [ 2 ,   2 ] −8.4984 QPR-T1-1 0.1 [ 1   1   1 ] 50 100 [ - 1 ,   - 1 ] [ 2 ,   2 ] 1For solving the first six problems, we have kept the initial guesses used by the IED in its previous version as described in [1]. For the rest of the problems, c0 and u0 were randomly chosen.One can notice from Tables2–21 that IED + GA has found the solution for all the problems regardless of the norms used whereas IED + fmin failed some of them.Table 2
Comparison of the results obtained by both versions of the IED for problem GLR-P1-1.
σ ( ⋅ ) GLR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −2.6273 ×104 - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 ⋅ 2 - 2.6273 × 1 0 4 - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 m ⋅ ∞ — - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 e ⋅ 2 - 1 - 2.6273 × 1 0 4 - 2.6273 × 1 0 4 - 2.6270 × 1 0 4 - 2.6272 × 1 0 4 0.8555 e ⋅ 1 - 1 - 2.6273 × 1 0 4 - 2.6273 × 1 0 4 - 2.6270 × 1 0 4 - 2.6272 × 1 0 4 0.8555 m(e⋅∞-1) — - 2.6272 × 1 0 4 - 2.6225 × 1 0 4 - 2.6261 × 1 0 4 14.5763 m a x ( ⋅ 2 , ⋅ 2 2 ) — - 2.6273 × 1 0 4 - 2.4727 × 1 0 4 - 2.6049 × 1 0 4 477.8463Table 3
Comparison of the results obtained by both versions of the IED for problem GQR-P1-1.
σ ( ⋅ ) GQR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −7.7978 −6.7495 −7.5575 0.4268 ⋅ 2 — −7.7978 −6.7495 −7.5572 0.4276 m ⋅ ∞ −7.8028 −7.7984 −6.7495 −7.0554 0.4927 e ⋅ 2 - 1 −7.8028 −7.7925 −6.7495 −7.4544 0.4872 e ⋅ 1 - 1 −7.8028 −7.8012 −7.7443 −7.7801 0.0171 m(e⋅∞-1) −7.8028 −7.7985 −6.7495 −7.4601 0.4914 m a x ( ⋅ 2 , ⋅ 2 2 ) −7.8028 −7.8024 −6.7495 −7.3560 0.5228Table 4
Comparison of the results obtained by both versions of the IED for problem PPR-P1-2.
σ ( ⋅ ) PPR-P1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 6.2998 × 1 0 3 6.3002 × 1 0 3 6.3018 × 1 0 3 6.3008 × 1 0 3 0.4452 ⋅ 2 6.2998 × 1 0 3 6.3002 × 1 0 3 6.3018 × 1 0 3 6.3008 × 1 0 3 0.4452 m ⋅ ∞ 6.2998 × 1 0 3 6.3001 × 1 0 3 6.3035 × 1 0 3 6.3015 × 1 0 3 1.0803 e ⋅ 2 - 1 — 6.3000 × 1 0 3 6.3023 × 1 0 3 6.3007 × 1 0 3 0.7107 e ⋅ 1 - 1 — 6.3000 × 1 0 3 6.3023 × 1 0 3 6.3007 × 1 0 3 0.7107 m(e⋅∞-1) — 6.3000 × 1 0 3 6.3053 × 1 0 3 6.3012 × 1 0 3 1.6442 m a x ( ⋅ 2 , ⋅ 2 2 ) 6.2999 × 1 0 3 6.2999 × 1 0 3 6.3045 × 1 0 3 6.3013 × 1 0 3 1.5406Table 5
Comparison of the results obtained by both versions of the IED for problem PQR-T1-7.
σ ( ⋅ ) PQR-T1-7 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −22.5467 −22.1224 −22.4160 0.1450 ⋅ 2 — −22.6033 −22.1466 −22.4274 0.1383 m ⋅ ∞ −22.6274 −22.6233 −22.3691 −22.5153 0.1151 e ⋅ 2 - 1 −22.6262 −22.5424 −22.0668 −22.3474 0.1526 e ⋅ 1 - 1 −22.6270 −22.5468 −21.9238 −22.2471 0.1891 m(e⋅∞-1) −22.6274 −22.5215 21.6514 −17.8111 13.8676 m a x ( ⋅ 2 , ⋅ 2 2 ) −22.6156 −22.5704 −22.0331 −22.3381 0.1683Table 6
Comparison of the results obtained by both versions of the IED for problem SQR-P1-1.
σ ( ⋅ ) SQR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.0285 0.0285 0.0287 0.0286 8.4993 × 1 0 - 5 ⋅ 2 0.0285 0.0285 0.0287 0.0286 8.2863 × 1 0 - 5 m ⋅ ∞ 0.0285 0.0285 0.0293 0.0287 2.3577 × 1 0 - 4 e ⋅ 2 - 1 0.0285 0.0285 0.0293 0.0287 5.7703 × 1 0 - 4 e ⋅ 1 - 1 0.0285 0.0285 0.0306 0.0289 6.5958 × 1 0 - 4 m(e⋅∞-1) 0.0285 0.0285 0.0290 0.0286 1.8364 × 1 0 - 4 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.0285 0.0285 0.0289 0.0286 1.2819 × 1 0 - 4Table 7
Comparison of the results obtained by both versions of the IED for problem LGR-P1-1.
σ ( ⋅ ) LGR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.5187 0.5234 0.5407 0.5314 0.0061 ⋅ 2 — 0.5216 0.5427 0.5301 0.0083 m ⋅ ∞ 0.5228 0.5213 0.5726 0.5364 0.0157 e ⋅ 2 - 1 0.5199 0.5213 0.5726 0.5364 0.0157 e ⋅ 1 - 1 0.5195 0.5232 0.5622 0.5384 0.0125 m(e⋅∞-1) 0.51 83 0.5239 0.5677 0.5390 0.0161 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.5182 0.5201 0.5488 0.5313 0.0090Table 8
Comparison of the results obtained by both versions of the IED for problem QBR-T1-1.
σ ( ⋅ ) QBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 ⋅ 2 — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 m ⋅ ∞ — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 e ⋅ 2 - 1 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 e ⋅ 1 - 1 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 m(e⋅∞-1) 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 m a x ( ⋅ 2 , ⋅ 2 2 ) 2.4563 × 1 0 - 14 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5Table 9
Comparison of the results obtained by both versions of the IED for problem PBR-T1-1.
σ ( ⋅ ) PBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 ⋅ 2 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m ⋅ ∞ - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 e ⋅ 2 - 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 e ⋅ 1 - 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m(e⋅∞-1) - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m a x ( ⋅ 2 , ⋅ 2 2 ) - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052Table 10
Comparison of the results obtained by both versions of the IED for problem PBR-T1-2.
σ ( ⋅ ) PBR-T1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.0504 0.0506 0.0677 0.0536 0.0052 ⋅ 2 0.0504 0.0506 0.0677 0.0536 0.0052 m ⋅ ∞ 0.0504 0.0506 0.0677 0.0536 0.0052 e ⋅ 2 - 1 0.0504 0.0508 0.0839 0.0593 0.0123 e ⋅ 1 - 1 0.0504 0.0508 0.0839 0.0593 0.0123 m(e⋅∞-1) 0.0504 0.0508 0.0839 0.0593 0.0123 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.0504 0.0505 0.0790 0.0545 0.0088Table 11
Comparison of the results obtained by both versions of the IED for problem QQR-P1-3.
σ ( ⋅ ) QQR-P1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.9535 0.9576 0.9837 0.9676 0.0099 ⋅ 2 0.9535 0.9565 0.9853 0.9637 0.0890 m ⋅ ∞ 0.9535 0.9550 0.9881 0.9665 0.0115 e ⋅ 2 - 1 0.9535 0.9550 0.9684 0.9603 0.0039 e ⋅ 1 - 1 0.9535 0.9541 0.9860 0.9652 0.0096 m(e⋅∞-1) −25.0694 0.9558 0.9683 0.9628 0.0040 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.9535 0.9568 1.0124 0.9708 0.0185Table 12
Comparison of the results obtained by both versions of the IED for problem QQR-T1-6.
σ ( ⋅ ) QQR-T1-6 IED + fmin IED + GA result best worst mean sd ⋅ 1 1 1.0002 1.0185 1.0052 0.0057 ⋅ 2 1 1.0023 1.0179 1.0101 0.0044 m ⋅ ∞ 1 1.0003 1.0404 1.0112 0.0125 e ⋅ 2 - 1 1 1.0005 1.0287 1.0126 0.0093 e ⋅ 1 - 1 1 1.0005 1.0400 1.0122 0.0137 m(e⋅∞-1) 1 1.0002 1.0366 1.0133 0.0106 m a x ( ⋅ 2 , ⋅ 2 2 ) 1 1.0002 1.0097 1.0046 0.0037Table 13
Comparison of the results obtained by both versions of the IED for problem PLR-T1-1.
σ ( ⋅ ) PLR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −0.9976 −0.9729 −0.9915 0.0071 ⋅ 2 — −0.9966 −0.9817 −0.9892 0.0052 m ⋅ ∞ — −0.9984 −0.9663 −0.9899 0.0102 e ⋅ 2 - 1 −1 −0.9974 −0.9806 −0.9916 0.0057 e ⋅ 1 - 1 −1 −0.9966 −0.9805 −0.9925 0.0048 m(e⋅∞-1) −1 −0.9993 −0.9793 −0.9926 0.0061 m a x ( ⋅ 2 , ⋅ 2 2 ) −1 −0.9996 −0.9758 −0.9920 0.0073Table 14
Comparison of the results obtained by both versions of the IED for problem PBR-T1-3.
σ ( ⋅ ) PBR-T1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 2.6686 2.7135 2.6827 0.0134 ⋅ 2 — 2.6698 2.6855 2.6743 0.0050 m ⋅ ∞ — 2.6697 2.6909 2.6784 0.0079 e ⋅ 2 - 1 2.6667 2.6692 2.6947 2.6784 0.0082 e ⋅ 1 - 1 2.6667 2.6685 2.7022 2.6809 0.0110 m(e⋅∞-1) 2.6667 2.6687 2.7059 2.6909 0.0125 m a x ( ⋅ 2 , ⋅ 2 2 ) 2.6667 2.6684 2.7125 2.6847 0.0150Table 15
Comparison of the results obtained by both versions of the IED for problem QLR-T1-1.
σ ( ⋅ ) QLR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −99.9600 −99.9600 −99.9597 −99.9599 8.9350 × 1 0 - 5 ⋅ 2 −99.9600 −99.9600 −99.9595 −99.9598 1.4311 × 1 0 - 4 m ⋅ ∞ −99.9600 −99.9600 −99.9599 −99.9599 5.1418 × 1 0 - 5 e ⋅ 2 - 1 −99.9600 −99.9600 −99.9594 −99.9599 1.5289 × 1 0 - 4 e ⋅ 1 - 1 −99.9600 −99.9600 −99.9595 −99.9599 1.3253 × 1 0 - 4 m(e⋅∞-1) −99.9600 −99.9600 −99.9594 −99.9598 1.8280 × 1 0 - 4 m a x ( ⋅ 2 , ⋅ 2 2 ) −99.9600 −99.9599 −99.9598 −99.9599 3.7864 × 1 0 - 5Table 16
Comparison of the results obtained by both versions of the IED for problem PQR-T1-1.
σ ( ⋅ ) PQR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 306.5 306.5146 326.6933 310.9483 6.2614 ⋅ 2 306.5 308.7375 329.3517 312.3063 6.1269 m ⋅ ∞ 306.5 307.5067 327.2424 311.6510 5.8228 e ⋅ 2 - 1 306.5 308.9435 317.7634 312.9006 2.7421 e ⋅ 1 - 1 306.5 306.5408 315.6229 310.7872 3.0505 m(e⋅∞-1) 306.5 307.8027 316.7068 312.1473 2.7962 m a x ( ⋅ 2 , ⋅ 2 2 ) 306.5 307.9935 315.4329 311.4265 2.4128Table 17
Comparison of the results obtained by both versions of the IED for problem QQR-T1-3.
σ ( ⋅ ) QQR-T1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −29.9998 −29.8286 −29.9714 0.0532 ⋅ 2 — −29.9998 −29.8286 −29.9714 0.532 m ⋅ ∞ — −29.9998 −29.8286 −29.9714 0.0532 e ⋅ 2 - 1 −30 −29.9972 −29.3533 −29.8566 0.2162 e ⋅ 1 - 1 −30 −29.9972 −29.3533 −29.8566 0.2162 m(e⋅∞-1) −30 −29.9972 −29.3533 −29.8566 0.2162 m a x ( ⋅ 2 , ⋅ 2 2 ) −30 −29.9960 −29.2340 −29.8780 0.2316Table 18
Comparison of the results obtained by both versions of the IED for problem GBR-T1-1.
σ ( ⋅ ) GBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7 ⋅ 2 −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7 m ⋅ ∞ −1.9132 −1.9132 −1.9132 −1.9132 1.4981 × 1 0 - 6 e ⋅ 2 - 1 −1.9132 −1.9132 −1.9132 −1.9132 9.3127 × 1 0 - 7 e ⋅ 1 - 1 −1.9132 −1.9132 −1.9132 −1.9132 9.3127 × 1 0 - 7 m(e⋅∞-1) −1.9132 −1.9132 −1.9132 −1.9132 1.2166 × 1 0 - 6 m a x ( ⋅ 2 , ⋅ 2 2 ) −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7Table 19
Comparison of the results obtained by both versions of the IED for problem PQR-T1-4.
σ ( ⋅ ) PQR-T1-4 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −6791.5 173850 1237.6 5.6747 × 1 0 4 ⋅ 2 — −6936.9 1792.1 −5153.5 2.8982 × 1 0 3 m ⋅ ∞ — −6800.2 −2548.9 −5962.7 1.3694 × 1 0 3 e ⋅ 2 - 1 −6961.8 −6874.3 −6258.0 −6737.2 188.5621 e ⋅ 1 - 1 −6961.8 −6915.8 −1642.2 −6165.1 1606.1 m(e⋅∞-1) −6961.8 −6777.9 −6159.4 −6590.4 222.8402 m a x ( ⋅ 2 , ⋅ 2 2 ) −6961.8 −6958.0 −1735.3 −6274.8 1609.0Table 20
Comparison of the results obtained by both versions of the IED for problem QQR-T1-2.
σ ( ⋅ ) QQR-T1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 −8.4985 −8.4958 −8.3391 −8.4729 0.0481 ⋅ 2 −8.4985 −8.4958 −8.3391 −8.4729 0.0481 m ⋅ ∞ −8.4985 −8.4958 −8.3391 −8.4729 0.0481 e ⋅ 2 - 1 −8.4985 −8.4958 −7.5694 −8.2675 0.3201 e ⋅ 1 - 1 −8.4985 −8.4958 −7.5694 −8.2675 0.3201 m(e⋅∞-1) −8.4985 −8.4958 −7.5694 −8.2675 0.3201 m a x ( ⋅ 2 , ⋅ 2 2 ) −8.4985 −8.4980 −8.0434 −8.3737 0.1564Table 21
Comparison of the results obtained by both versions of the IED for problem QPR-T1-1.
σ ( ⋅ ) QPR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 1.1349 1.4578 1.2706 0.0846 ⋅ 2 — 1.0632 1.4943 1.2132 0.1356 m ⋅ ∞ — 1.1317 1.6091 1.2670 0.1421 e ⋅ 2 - 1 0.9978 1.1316 1.4839 1.2715 0.1220 e ⋅ 1 - 1 — 1.0729 1.5440 1.2488 0.1375 m(e⋅∞-1) 0.9979 1.0818 1.4799 1.2300 0.1555 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.9987 1.0967 1.6843 1.2762 0.1853It is interesting to highlight that the standard deviation of all the problems is small. This shows robustness of the version of the IED method proposed in this work.Tables2–21 show the results of the problems obtained by both versions of the algorithm. For the IED + GA algorithm, we present the best and the worst results, the mean and the standard deviation. Cases where a solution has been not found are indicated by a line struck through the entry of the table.


## 7.1.1. Comparison of Results of the First Set of Numerical Experiments

Table1 provides the initial guesses c0 and u0 used by IED + fmin and IED + GA, the size of the population, the number of generations, and bound constraints used by the Genetic Algorithm. The table also shows the optimal values.Table 1
Parameters of the IED method.
Test-problem c 0 u 0 Population Size Generations Lower Bound Upper Bound Optimal Value GLR-P1-1 50 [ 0   0   0   0 ] 50 300 [ 0.01 ,   0.01 ,   0.01 ] [ 1 ,   1 ,   1 ] -2.6273×104 GQR-P1-1 1 [ 1   1   1   1   1   1   1 ] 50 100 [ 10 ,   10 ] [ 60 ,   60 ] -7.8028 PPR-P1-2 0.1 [ 1   1   1   1 ] 50 200 [ 0 ,   0 ,   0 ] [ 300 ,   300 ,   300 ] 6.2998×103 PQR-T1-7 1.5 [ 1   1   1   1   1   1   1 ] 50 100 [ - 5 ,   - 5 ,   - 5 ] [ 5 ,   5 ,   5 ] -22.6274 SQR-P1-1 0.2 [ 0   0   0 ] 50 100 [ - 10 ,   - 10 ] [ 10 ,   10 ] 0.0285 LGR-P1-1 0.1 [ 0   0   0   0   0   0   0   0 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0.5181 QBR-T1-1 1 [ 1   1   1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0 PBR-T1-1 1 [ 1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 0 PBR-T1-2 1 [ 1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 0.0504 QQR-P1-3 1 [ 1   1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ,   0 ] [ 5 ,   5 ,   5 ] 0.9535 QQR-T1-6 1 [ 1   1 ] 50 100 [ - 2 ,   - 2 ] [ 2 ,   2 ] 1 PLR-T1-1 1 [ 1   1   1   1   1 ] 50 100 [ 1 ,   1 ] [ 4 ,   4 ] −1 PBR-T1-3 1 [ 1   1 ] 50 100 [ - 1 ,   - 1 ] [ 2 ,   2 ] 2.6666 QLR-T1-1 1 [ 1   1   1   1   1 ] 50 100 [ - 1 ,   - 1 ] [ 3 ,   3 ] −99.96 PQR-T1-1 5 [ 1   1   1 ] 50 100 [ 0 ,   0 ] [ 3 ,   3 ] 306.5 QQR-T1-3 0.1 [ 1 ] 20 50 [ 1 ,   1 ] [ 4 ,   4 ] −30 GBR-T1-1 1 [ 1   1   1   1 ] 50 100 [ - 2 ,   - 2 ] [ 0 ,   0 ] −1.9132 PQR-T1-4 1 [ 1   1   1   1   1   1 ] 50 100 [ 0 ,   0 ] [ 15 ,   15 ] −6961.8138 QQR-T1-2 1 [ 1 ] 20 50 [ 0 ,   0 ] [ 2 ,   2 ] −8.4984 QPR-T1-1 0.1 [ 1   1   1 ] 50 100 [ - 1 ,   - 1 ] [ 2 ,   2 ] 1For solving the first six problems, we have kept the initial guesses used by the IED in its previous version as described in [1]. For the rest of the problems, c0 and u0 were randomly chosen.One can notice from Tables2–21 that IED + GA has found the solution for all the problems regardless of the norms used whereas IED + fmin failed some of them.Table 2
Comparison of the results obtained by both versions of the IED for problem GLR-P1-1.
σ ( ⋅ ) GLR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −2.6273 ×104 - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 ⋅ 2 - 2.6273 × 1 0 4 - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 m ⋅ ∞ — - 2.6272 × 1 0 4 - 2.6267 × 1 0 4 - 2.6271 × 1 0 4 2.0661 e ⋅ 2 - 1 - 2.6273 × 1 0 4 - 2.6273 × 1 0 4 - 2.6270 × 1 0 4 - 2.6272 × 1 0 4 0.8555 e ⋅ 1 - 1 - 2.6273 × 1 0 4 - 2.6273 × 1 0 4 - 2.6270 × 1 0 4 - 2.6272 × 1 0 4 0.8555 m(e⋅∞-1) — - 2.6272 × 1 0 4 - 2.6225 × 1 0 4 - 2.6261 × 1 0 4 14.5763 m a x ( ⋅ 2 , ⋅ 2 2 ) — - 2.6273 × 1 0 4 - 2.4727 × 1 0 4 - 2.6049 × 1 0 4 477.8463Table 3
Comparison of the results obtained by both versions of the IED for problem GQR-P1-1.
σ ( ⋅ ) GQR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −7.7978 −6.7495 −7.5575 0.4268 ⋅ 2 — −7.7978 −6.7495 −7.5572 0.4276 m ⋅ ∞ −7.8028 −7.7984 −6.7495 −7.0554 0.4927 e ⋅ 2 - 1 −7.8028 −7.7925 −6.7495 −7.4544 0.4872 e ⋅ 1 - 1 −7.8028 −7.8012 −7.7443 −7.7801 0.0171 m(e⋅∞-1) −7.8028 −7.7985 −6.7495 −7.4601 0.4914 m a x ( ⋅ 2 , ⋅ 2 2 ) −7.8028 −7.8024 −6.7495 −7.3560 0.5228Table 4
Comparison of the results obtained by both versions of the IED for problem PPR-P1-2.
σ ( ⋅ ) PPR-P1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 6.2998 × 1 0 3 6.3002 × 1 0 3 6.3018 × 1 0 3 6.3008 × 1 0 3 0.4452 ⋅ 2 6.2998 × 1 0 3 6.3002 × 1 0 3 6.3018 × 1 0 3 6.3008 × 1 0 3 0.4452 m ⋅ ∞ 6.2998 × 1 0 3 6.3001 × 1 0 3 6.3035 × 1 0 3 6.3015 × 1 0 3 1.0803 e ⋅ 2 - 1 — 6.3000 × 1 0 3 6.3023 × 1 0 3 6.3007 × 1 0 3 0.7107 e ⋅ 1 - 1 — 6.3000 × 1 0 3 6.3023 × 1 0 3 6.3007 × 1 0 3 0.7107 m(e⋅∞-1) — 6.3000 × 1 0 3 6.3053 × 1 0 3 6.3012 × 1 0 3 1.6442 m a x ( ⋅ 2 , ⋅ 2 2 ) 6.2999 × 1 0 3 6.2999 × 1 0 3 6.3045 × 1 0 3 6.3013 × 1 0 3 1.5406Table 5
Comparison of the results obtained by both versions of the IED for problem PQR-T1-7.
σ ( ⋅ ) PQR-T1-7 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −22.5467 −22.1224 −22.4160 0.1450 ⋅ 2 — −22.6033 −22.1466 −22.4274 0.1383 m ⋅ ∞ −22.6274 −22.6233 −22.3691 −22.5153 0.1151 e ⋅ 2 - 1 −22.6262 −22.5424 −22.0668 −22.3474 0.1526 e ⋅ 1 - 1 −22.6270 −22.5468 −21.9238 −22.2471 0.1891 m(e⋅∞-1) −22.6274 −22.5215 21.6514 −17.8111 13.8676 m a x ( ⋅ 2 , ⋅ 2 2 ) −22.6156 −22.5704 −22.0331 −22.3381 0.1683Table 6
Comparison of the results obtained by both versions of the IED for problem SQR-P1-1.
σ ( ⋅ ) SQR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.0285 0.0285 0.0287 0.0286 8.4993 × 1 0 - 5 ⋅ 2 0.0285 0.0285 0.0287 0.0286 8.2863 × 1 0 - 5 m ⋅ ∞ 0.0285 0.0285 0.0293 0.0287 2.3577 × 1 0 - 4 e ⋅ 2 - 1 0.0285 0.0285 0.0293 0.0287 5.7703 × 1 0 - 4 e ⋅ 1 - 1 0.0285 0.0285 0.0306 0.0289 6.5958 × 1 0 - 4 m(e⋅∞-1) 0.0285 0.0285 0.0290 0.0286 1.8364 × 1 0 - 4 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.0285 0.0285 0.0289 0.0286 1.2819 × 1 0 - 4Table 7
Comparison of the results obtained by both versions of the IED for problem LGR-P1-1.
σ ( ⋅ ) LGR-P1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.5187 0.5234 0.5407 0.5314 0.0061 ⋅ 2 — 0.5216 0.5427 0.5301 0.0083 m ⋅ ∞ 0.5228 0.5213 0.5726 0.5364 0.0157 e ⋅ 2 - 1 0.5199 0.5213 0.5726 0.5364 0.0157 e ⋅ 1 - 1 0.5195 0.5232 0.5622 0.5384 0.0125 m(e⋅∞-1) 0.51 83 0.5239 0.5677 0.5390 0.0161 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.5182 0.5201 0.5488 0.5313 0.0090Table 8
Comparison of the results obtained by both versions of the IED for problem QBR-T1-1.
σ ( ⋅ ) QBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 ⋅ 2 — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 m ⋅ ∞ — 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5 e ⋅ 2 - 1 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 e ⋅ 1 - 1 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 m(e⋅∞-1) 3.668 × 1 0 - 17 7.2718 × 1 0 - 6 2.0872 × 1 0 - 4 9.4788 × 1 0 - 5 8.2756 × 1 0 - 5 m a x ( ⋅ 2 , ⋅ 2 2 ) 2.4563 × 1 0 - 14 5.4134 × 1 0 - 7 1.5704 × 1 0 - 4 6.7404 × 1 0 - 5 5.3498 × 1 0 - 5Table 9
Comparison of the results obtained by both versions of the IED for problem PBR-T1-1.
σ ( ⋅ ) PBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 ⋅ 2 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m ⋅ ∞ - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 e ⋅ 2 - 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 e ⋅ 1 - 1 - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m(e⋅∞-1) - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052 m a x ( ⋅ 2 , ⋅ 2 2 ) - 9.0273 x 1 0 - 14 0 0.0134 0.0027 0.0052Table 10
Comparison of the results obtained by both versions of the IED for problem PBR-T1-2.
σ ( ⋅ ) PBR-T1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.0504 0.0506 0.0677 0.0536 0.0052 ⋅ 2 0.0504 0.0506 0.0677 0.0536 0.0052 m ⋅ ∞ 0.0504 0.0506 0.0677 0.0536 0.0052 e ⋅ 2 - 1 0.0504 0.0508 0.0839 0.0593 0.0123 e ⋅ 1 - 1 0.0504 0.0508 0.0839 0.0593 0.0123 m(e⋅∞-1) 0.0504 0.0508 0.0839 0.0593 0.0123 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.0504 0.0505 0.0790 0.0545 0.0088Table 11
Comparison of the results obtained by both versions of the IED for problem QQR-P1-3.
σ ( ⋅ ) QQR-P1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 0.9535 0.9576 0.9837 0.9676 0.0099 ⋅ 2 0.9535 0.9565 0.9853 0.9637 0.0890 m ⋅ ∞ 0.9535 0.9550 0.9881 0.9665 0.0115 e ⋅ 2 - 1 0.9535 0.9550 0.9684 0.9603 0.0039 e ⋅ 1 - 1 0.9535 0.9541 0.9860 0.9652 0.0096 m(e⋅∞-1) −25.0694 0.9558 0.9683 0.9628 0.0040 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.9535 0.9568 1.0124 0.9708 0.0185Table 12
Comparison of the results obtained by both versions of the IED for problem QQR-T1-6.
σ ( ⋅ ) QQR-T1-6 IED + fmin IED + GA result best worst mean sd ⋅ 1 1 1.0002 1.0185 1.0052 0.0057 ⋅ 2 1 1.0023 1.0179 1.0101 0.0044 m ⋅ ∞ 1 1.0003 1.0404 1.0112 0.0125 e ⋅ 2 - 1 1 1.0005 1.0287 1.0126 0.0093 e ⋅ 1 - 1 1 1.0005 1.0400 1.0122 0.0137 m(e⋅∞-1) 1 1.0002 1.0366 1.0133 0.0106 m a x ( ⋅ 2 , ⋅ 2 2 ) 1 1.0002 1.0097 1.0046 0.0037Table 13
Comparison of the results obtained by both versions of the IED for problem PLR-T1-1.
σ ( ⋅ ) PLR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −0.9976 −0.9729 −0.9915 0.0071 ⋅ 2 — −0.9966 −0.9817 −0.9892 0.0052 m ⋅ ∞ — −0.9984 −0.9663 −0.9899 0.0102 e ⋅ 2 - 1 −1 −0.9974 −0.9806 −0.9916 0.0057 e ⋅ 1 - 1 −1 −0.9966 −0.9805 −0.9925 0.0048 m(e⋅∞-1) −1 −0.9993 −0.9793 −0.9926 0.0061 m a x ( ⋅ 2 , ⋅ 2 2 ) −1 −0.9996 −0.9758 −0.9920 0.0073Table 14
Comparison of the results obtained by both versions of the IED for problem PBR-T1-3.
σ ( ⋅ ) PBR-T1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 2.6686 2.7135 2.6827 0.0134 ⋅ 2 — 2.6698 2.6855 2.6743 0.0050 m ⋅ ∞ — 2.6697 2.6909 2.6784 0.0079 e ⋅ 2 - 1 2.6667 2.6692 2.6947 2.6784 0.0082 e ⋅ 1 - 1 2.6667 2.6685 2.7022 2.6809 0.0110 m(e⋅∞-1) 2.6667 2.6687 2.7059 2.6909 0.0125 m a x ( ⋅ 2 , ⋅ 2 2 ) 2.6667 2.6684 2.7125 2.6847 0.0150Table 15
Comparison of the results obtained by both versions of the IED for problem QLR-T1-1.
σ ( ⋅ ) QLR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −99.9600 −99.9600 −99.9597 −99.9599 8.9350 × 1 0 - 5 ⋅ 2 −99.9600 −99.9600 −99.9595 −99.9598 1.4311 × 1 0 - 4 m ⋅ ∞ −99.9600 −99.9600 −99.9599 −99.9599 5.1418 × 1 0 - 5 e ⋅ 2 - 1 −99.9600 −99.9600 −99.9594 −99.9599 1.5289 × 1 0 - 4 e ⋅ 1 - 1 −99.9600 −99.9600 −99.9595 −99.9599 1.3253 × 1 0 - 4 m(e⋅∞-1) −99.9600 −99.9600 −99.9594 −99.9598 1.8280 × 1 0 - 4 m a x ( ⋅ 2 , ⋅ 2 2 ) −99.9600 −99.9599 −99.9598 −99.9599 3.7864 × 1 0 - 5Table 16
Comparison of the results obtained by both versions of the IED for problem PQR-T1-1.
σ ( ⋅ ) PQR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 306.5 306.5146 326.6933 310.9483 6.2614 ⋅ 2 306.5 308.7375 329.3517 312.3063 6.1269 m ⋅ ∞ 306.5 307.5067 327.2424 311.6510 5.8228 e ⋅ 2 - 1 306.5 308.9435 317.7634 312.9006 2.7421 e ⋅ 1 - 1 306.5 306.5408 315.6229 310.7872 3.0505 m(e⋅∞-1) 306.5 307.8027 316.7068 312.1473 2.7962 m a x ( ⋅ 2 , ⋅ 2 2 ) 306.5 307.9935 315.4329 311.4265 2.4128Table 17
Comparison of the results obtained by both versions of the IED for problem QQR-T1-3.
σ ( ⋅ ) QQR-T1-3 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −29.9998 −29.8286 −29.9714 0.0532 ⋅ 2 — −29.9998 −29.8286 −29.9714 0.532 m ⋅ ∞ — −29.9998 −29.8286 −29.9714 0.0532 e ⋅ 2 - 1 −30 −29.9972 −29.3533 −29.8566 0.2162 e ⋅ 1 - 1 −30 −29.9972 −29.3533 −29.8566 0.2162 m(e⋅∞-1) −30 −29.9972 −29.3533 −29.8566 0.2162 m a x ( ⋅ 2 , ⋅ 2 2 ) −30 −29.9960 −29.2340 −29.8780 0.2316Table 18
Comparison of the results obtained by both versions of the IED for problem GBR-T1-1.
σ ( ⋅ ) GBR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7 ⋅ 2 −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7 m ⋅ ∞ −1.9132 −1.9132 −1.9132 −1.9132 1.4981 × 1 0 - 6 e ⋅ 2 - 1 −1.9132 −1.9132 −1.9132 −1.9132 9.3127 × 1 0 - 7 e ⋅ 1 - 1 −1.9132 −1.9132 −1.9132 −1.9132 9.3127 × 1 0 - 7 m(e⋅∞-1) −1.9132 −1.9132 −1.9132 −1.9132 1.2166 × 1 0 - 6 m a x ( ⋅ 2 , ⋅ 2 2 ) −1.9132 −1.9132 −1.9132 −1.9132 5.1999 × 1 0 - 7Table 19
Comparison of the results obtained by both versions of the IED for problem PQR-T1-4.
σ ( ⋅ ) PQR-T1-4 IED + fmin IED + GA result best worst mean sd ⋅ 1 — −6791.5 173850 1237.6 5.6747 × 1 0 4 ⋅ 2 — −6936.9 1792.1 −5153.5 2.8982 × 1 0 3 m ⋅ ∞ — −6800.2 −2548.9 −5962.7 1.3694 × 1 0 3 e ⋅ 2 - 1 −6961.8 −6874.3 −6258.0 −6737.2 188.5621 e ⋅ 1 - 1 −6961.8 −6915.8 −1642.2 −6165.1 1606.1 m(e⋅∞-1) −6961.8 −6777.9 −6159.4 −6590.4 222.8402 m a x ( ⋅ 2 , ⋅ 2 2 ) −6961.8 −6958.0 −1735.3 −6274.8 1609.0Table 20
Comparison of the results obtained by both versions of the IED for problem QQR-T1-2.
σ ( ⋅ ) QQR-T1-2 IED + fmin IED + GA result best worst mean sd ⋅ 1 −8.4985 −8.4958 −8.3391 −8.4729 0.0481 ⋅ 2 −8.4985 −8.4958 −8.3391 −8.4729 0.0481 m ⋅ ∞ −8.4985 −8.4958 −8.3391 −8.4729 0.0481 e ⋅ 2 - 1 −8.4985 −8.4958 −7.5694 −8.2675 0.3201 e ⋅ 1 - 1 −8.4985 −8.4958 −7.5694 −8.2675 0.3201 m(e⋅∞-1) −8.4985 −8.4958 −7.5694 −8.2675 0.3201 m a x ( ⋅ 2 , ⋅ 2 2 ) −8.4985 −8.4980 −8.0434 −8.3737 0.1564Table 21
Comparison of the results obtained by both versions of the IED for problem QPR-T1-1.
σ ( ⋅ ) QPR-T1-1 IED + fmin IED + GA result best worst mean sd ⋅ 1 — 1.1349 1.4578 1.2706 0.0846 ⋅ 2 — 1.0632 1.4943 1.2132 0.1356 m ⋅ ∞ — 1.1317 1.6091 1.2670 0.1421 e ⋅ 2 - 1 0.9978 1.1316 1.4839 1.2715 0.1220 e ⋅ 1 - 1 — 1.0729 1.5440 1.2488 0.1375 m(e⋅∞-1) 0.9979 1.0818 1.4799 1.2300 0.1555 m a x ( ⋅ 2 , ⋅ 2 2 ) 0.9987 1.0967 1.6843 1.2762 0.1853It is interesting to highlight that the standard deviation of all the problems is small. This shows robustness of the version of the IED method proposed in this work.Tables2–21 show the results of the problems obtained by both versions of the algorithm. For the IED + GA algorithm, we present the best and the worst results, the mean and the standard deviation. Cases where a solution has been not found are indicated by a line struck through the entry of the table.


## 7.2. The Second Set of Numerical Experiments

In this section the algorithm proposed is applied to solve 4 real-world mechanical engineering optimization problems, largely used as a test-bed in the literature.The Tension/Compression String Design. In this problem, the objective is to minimize the volume V of a coil spring, depicted in the Figure 8, under a constant tension/compression load. There are three design variables to be considered in this problem: The number x1=N of active coils of the spring, the winding diameter x2=D and the wire diameter x3=d. The volume of the coil to be minimized is written as [27] (74)Vx=x1+2x2x32and is subject to the constraints (75)g1x=1-x23x171785x34≤0g2x=4x22-x3x212566x2x33-x34+15108x32-1≤0g3x=1-140.45x3x22x1≤0g4x=x2+x31.5-1≤0,where (76)2≤x1≤150.25≤x2≤1.30.05≤x3≤2.The Speed Reducer Design. The objective of this problem is to minimize the weight W of the speed reducer [27] shown in the Figure 9. The design variables are the face width (x1=b), the module of teeth (x2=m), the number of teeth on pinion (x3=n), the length of the shaft 1 between the bearings (x4=l1), the length of the shaft 2 between the bearings (x5=l2), the diameter of the shaft 1 (x6=d1), and, finally, the diameter of the shaft 2 (x7=d2). The third variable is integer and all the others are continuous. The constraints include limitations on the bending and surface stress of the gear teeth, transverse deflections of the shafts 1 and 2 generated by the transmitted force, and, finally, the stress in the shafts 1 and 2. The weight of the speed reducer, to be minimized, is given by(77)Wx=0.7854x1x223.3333x32+14.9334x3-43.0934-1.508x1x62+x72+7.477x63+x73+0.7854x4x62+x5x72subject to (78)g1x=27x1-1x2-2x3-1≤1g2x=397.5x1-1x2-2x3-2≤1g3x=1.93x2-1x3-1x43x6-4≤1g4x=1.93x2-1x3-1x53x7-4≤1g5x=745x4/x2x32+16.9×1060.50.1x63≤1100g6x=745x5/x2x32+157.5×1060.50.1x73≤850g7x=x2x3≤40g8x=x1x2≥5g9x=x1x2≤12g10x=1.5x6+1.9x4-1≤1g11x=1.1x7+1.9x5-1≤12.6≤x1≤3.60.7≤x2≤0.817≤x3≤287.3≤x4≤8.37.8≤x5≤8.32.9≤x6≤3.95.0≤x6≤5.5.Figure 9
The Speed Reducer design.The Welded Beam. This problem corresponds to the design of the welded beam [27] depicted in the Figure 10. The design variables are {h,l,t,b}, with bounds 0.125≤h≤10, and 0.1≤l,t,b≤10. The objective function to be minimized is the cost of the beam given as (79)Ch,l,t,b=1.10471h2l+0.04811tb14.0+lsubject to (80)g1τ=1-τ13600≥0g2σ=1-σ30000≥0g3b,h=b-h≥0g4Pc=Pc6000-1≥0g5δ=0.25-δ≥0.The expressions for τ, σ, Pc, and δ involve the design variables and are given by (81)τ=τ′2+τ′′2+lτ′τ′′αα=0.25l2+h+t2,σ=504000t2bPc=64746.0221-0.0282346ttb3δ=2.1952t3b,τ′=60002hlτ′′=600014+0.5lα20.707hll2/12+0.25h+t2.Figure 10
The welded beam.The Pressure Vessel. This problem corresponds to the weight minimization of a cylindrical pressure vessel with two spherical heads [27] as shown in Figure 11. The objective function involves four variables: the thickness of the pressure vessel (Ts), the thickness of the head (Th), the inner radius of the vessel (R), and the length of the cylindrical component (L). Since there are two discrete variables (Ts and Th) and two continuous variables (R and L), one has a nonlinearly constrained mixed discrete-continuous optimization problem.Figure 11
The pressure vessel.The bounds of the design variables are0,0625≤Ts,Th≤5 (in constant steps of 0.0625) and 10≤R,L≤200. The design variables are given in inches and the weight is written as (82)wTs,Th,R,L=0,6224TsThRL+1.7781ThR2+3.1661Ts2L+19.84Ts2Rto be minimized subject to the constraints (83)g1Ts,R=Ts-0.0193R≥0g2Th,R=Th-0.00954R≥0g3R,L=πR2L+4/3πR31296000-1≥0g4L=-L+240≥0.The first two constraints establish a lower bound to the ratios Ts/R and Th/R, respectively. The third constraint corresponds to a lower bound for the volume of the vessel and the last one to an upper bound to length of the cylindrical component.


### 7.2.1. Comparison of Results of the Second Set of Numerical Experiments

Table22 provides the initial guesses c0 and u0 used by IED + GA, the size of the population, and the number of generations used by the Genetic Algorithm.Table 22
Parameters of the IED method.
Problems c 0 u 0 Population Size Generations The Tension/Compression String 10 [ 1   1   1   1 ] 100 300 The Speed Reducer 1 [ 1   1   1   1   1   1   1   1   1   1   1 ] 100 300 The Welded Beam 10 [ 1   1   1   1   1 ] 100 300 The Pressure Vessel 1 [ 1   1   1   1 ] 100 300Table23 presents the best and the worst results, the mean, and the standard deviation. Tables 24–27 show the results found by the algorithm.Table 23
Statistical Analysis obtained by IED + GA in the engineering problems.
Engineering problems IED + GA best worst mean sd The Tension/Compression String 0.0128 0.0164 0.0144 0.0009 The Speed Reducer 2996.7 3039.7 3015.0 13.7485 The Welded Beam 2.4151 5.4177 3.1995 0.7005 The Pressure Vessel 6477.2 8787.4 7454.1 636.1933Table 24
Comparison for The Tension/Compression String problem.
Var. Alg-1 [11] Alg-2 [11] IED + GA x 1 11.852177 11.329555 11.184056 x 2 0.34747463 0.35603234 0.36004951 x 3 0.051301897 0.051660806 0.051889349 g 1 - 0.00000012 - 0.000006437 - 0.003088921 g 2 - 0.00000047 - 0.000013709 - 0.003358194 g 3 - 4.03513200 - 4.052324300 - 4.026621039 g 4 - 0.73414900 - 0.728204600 - 0.725374090  V 0.012668 0.012666 0.012781Table 25
Comparison for the speed reducer problem.
Var. Ref. [12] Ref. [13] Alg-1 [11] Alg-2 [11] IED + GA x 1 3.506163 3.500000 3.500001 3.5 3.499684 x 2 0.700831 0.700000 0.700000 0.7 0.700011 x 3 17.0 17.0 17.0 17.0 17.290709 x 4 7.460181 7.300008 7.300017 7.3000035 7.302923 x 5 7.962143 7.715322 7.715326 7.7153225 7.806459 x 6 3.362900 3.350215 3.350216 3.3502147 3.350185 x 7 5.308949 5.286655 5.286654 5.2866545 5.286355 g 1 - 0.077734 - 0.07391524 - 0.07391554 - 0.07391524 - 0.07386154 g 2 - 0.201305 - 0.19799852 - 0.19799876 - 0.19799852 - 0.19795199 g 3 - 0.474119 - 0.49917084 - 0.49916983 - 0.49917156 - 0.49856110 g 4 - 0.897068 - 0.90464383 - 0.90464365 - 0.90464383 - 0.90120377 g 5 - 0.011021 - 0.00000023 - 0.00000155 - 0.000000119 0.000030538 g 6 - 0.012500 - 0.00000029 0.00000000 0.00000000 0.000187373 g 7 - 0.702147 - 0.70250000 - 0.70250000 - 0.702500000 - 0.702495228 g 8 - 0.000573 0.00000000 - 0.00000029 0.00000000 0.00010613 g 9 - 0.583095 - 0.5833333 - 0.58333320 - 0.583333300 - 0.583377553 g 10 - 0.069144 - 0.051326692 - 0.05132753 - 0.051326156 - 0.051711423 g 11 - 0.027920 - 0.00000018 - 0.00000077 - 0.000000357 - 0.011717075  V 3025.0051 2994.4717 2994.4720 2994.4712 2996.7085Table 26
Comparison for the welded beam problem.
Var. Ref. [14] Alg-1 [11] Alg-2 Ref. [11] IED + GA x1 0.2442949 0.24432427 0.24438575 0.240091515 x2 6.2116738 6.2201996 6.2183037 6.390631032 x3 8.3015486 8.291464 8.291165 8.314142911 x4 0.2443003 0.24436942 0.24438748 0.246214266 g1 0.0004447 0.000000000 0.001953125 0.004494538 g2 64.378068 0.001953125 0.056640625 0.012900125 g3 0.0000054 0.000045150 0.000001728 0.006122751 g4 0.0002553 0.029785156 1.210937500 0.024765021 g5 0.2342937 0.234240830 0.234240280 0.234486548  V 2.38159 2.381246 2.3812175 2.4151069Table 27
Comparison for the pressure vessel problem.
Var. Ref. [13] Ref. [14] Alg-1 [11] Alg-2 [11] IED + GA Ts 0.8125 0.8125 0.8125 0.8125 0.90625 Th 0.4375 0.4375 0.4375 0.4375 0.46250 R 42.086994 42.0946558 42.093082 42.094967 46.189759 L 176.779128 176.684062 176.70308 176.67972 138.552311 g1 0.000221 0.000073 0.0001035 0.000007 0.014787 g2 0.035990 0.035917 0.0359320 0.035914 0.021849 g3 3.219817 2.929000 0.1562500 0.0625 0.035064 g4 63.220872 63.315938 63.296920 63.320282 101.447688  V 6061.1229 6060.187934 6060.3677 6060.138 6477.182


## 7.2.1. Comparison of Results of the Second Set of Numerical Experiments

Table22 provides the initial guesses c0 and u0 used by IED + GA, the size of the population, and the number of generations used by the Genetic Algorithm.Table 22
Parameters of the IED method.
Problems c 0 u 0 Population Size Generations The Tension/Compression String 10 [ 1   1   1   1 ] 100 300 The Speed Reducer 1 [ 1   1   1   1   1   1   1   1   1   1   1 ] 100 300 The Welded Beam 10 [ 1   1   1   1   1 ] 100 300 The Pressure Vessel 1 [ 1   1   1   1 ] 100 300Table23 presents the best and the worst results, the mean, and the standard deviation. Tables 24–27 show the results found by the algorithm.Table 23
Statistical Analysis obtained by IED + GA in the engineering problems.
Engineering problems IED + GA best worst mean sd The Tension/Compression String 0.0128 0.0164 0.0144 0.0009 The Speed Reducer 2996.7 3039.7 3015.0 13.7485 The Welded Beam 2.4151 5.4177 3.1995 0.7005 The Pressure Vessel 6477.2 8787.4 7454.1 636.1933Table 24
Comparison for The Tension/Compression String problem.
Var. Alg-1 [11] Alg-2 [11] IED + GA x 1 11.852177 11.329555 11.184056 x 2 0.34747463 0.35603234 0.36004951 x 3 0.051301897 0.051660806 0.051889349 g 1 - 0.00000012 - 0.000006437 - 0.003088921 g 2 - 0.00000047 - 0.000013709 - 0.003358194 g 3 - 4.03513200 - 4.052324300 - 4.026621039 g 4 - 0.73414900 - 0.728204600 - 0.725374090  V 0.012668 0.012666 0.012781Table 25
Comparison for the speed reducer problem.
Var. Ref. [12] Ref. [13] Alg-1 [11] Alg-2 [11] IED + GA x 1 3.506163 3.500000 3.500001 3.5 3.499684 x 2 0.700831 0.700000 0.700000 0.7 0.700011 x 3 17.0 17.0 17.0 17.0 17.290709 x 4 7.460181 7.300008 7.300017 7.3000035 7.302923 x 5 7.962143 7.715322 7.715326 7.7153225 7.806459 x 6 3.362900 3.350215 3.350216 3.3502147 3.350185 x 7 5.308949 5.286655 5.286654 5.2866545 5.286355 g 1 - 0.077734 - 0.07391524 - 0.07391554 - 0.07391524 - 0.07386154 g 2 - 0.201305 - 0.19799852 - 0.19799876 - 0.19799852 - 0.19795199 g 3 - 0.474119 - 0.49917084 - 0.49916983 - 0.49917156 - 0.49856110 g 4 - 0.897068 - 0.90464383 - 0.90464365 - 0.90464383 - 0.90120377 g 5 - 0.011021 - 0.00000023 - 0.00000155 - 0.000000119 0.000030538 g 6 - 0.012500 - 0.00000029 0.00000000 0.00000000 0.000187373 g 7 - 0.702147 - 0.70250000 - 0.70250000 - 0.702500000 - 0.702495228 g 8 - 0.000573 0.00000000 - 0.00000029 0.00000000 0.00010613 g 9 - 0.583095 - 0.5833333 - 0.58333320 - 0.583333300 - 0.583377553 g 10 - 0.069144 - 0.051326692 - 0.05132753 - 0.051326156 - 0.051711423 g 11 - 0.027920 - 0.00000018 - 0.00000077 - 0.000000357 - 0.011717075  V 3025.0051 2994.4717 2994.4720 2994.4712 2996.7085Table 26
Comparison for the welded beam problem.
Var. Ref. [14] Alg-1 [11] Alg-2 Ref. [11] IED + GA x1 0.2442949 0.24432427 0.24438575 0.240091515 x2 6.2116738 6.2201996 6.2183037 6.390631032 x3 8.3015486 8.291464 8.291165 8.314142911 x4 0.2443003 0.24436942 0.24438748 0.246214266 g1 0.0004447 0.000000000 0.001953125 0.004494538 g2 64.378068 0.001953125 0.056640625 0.012900125 g3 0.0000054 0.000045150 0.000001728 0.006122751 g4 0.0002553 0.029785156 1.210937500 0.024765021 g5 0.2342937 0.234240830 0.234240280 0.234486548  V 2.38159 2.381246 2.3812175 2.4151069Table 27
Comparison for the pressure vessel problem.
Var. Ref. [13] Ref. [14] Alg-1 [11] Alg-2 [11] IED + GA Ts 0.8125 0.8125 0.8125 0.8125 0.90625 Th 0.4375 0.4375 0.4375 0.4375 0.46250 R 42.086994 42.0946558 42.093082 42.094967 46.189759 L 176.779128 176.684062 176.70308 176.67972 138.552311 g1 0.000221 0.000073 0.0001035 0.000007 0.014787 g2 0.035990 0.035917 0.0359320 0.035914 0.021849 g3 3.219817 2.929000 0.1562500 0.0625 0.035064 g4 63.220872 63.315938 63.296920 63.320282 101.447688  V 6061.1229 6060.187934 6060.3677 6060.138 6477.182


## 8. Conclusions

We have proposed a version of the IED method for nonsmooth and nonconvex optimization problems that employs a Genetic Algorithm for minimization the Lagrangian function.We have solved twenty test problems for comparison between the first version and the current version to verify the effectiveness of the new approach proposed.We also solved four classical engineering problems to show that the method can be applied to more complex problems.We have evidence that shows that the more effectively we minimize the Lagrangian the better the IED method will be. Therefore, more powerful methods must be sought for this task. We also believe that the IED method can be bettered by establishing rules to update the matricesBk and employing different types of augmented Lagrangians with different matrices A and different functions σ.Finally, the hybridization presented in this work combines two algorithms, namely, the IED and a Genetic Algorithm. This approach frees the Genetic Algorithm of handling with constraints which is troublesome for these type of methods and, at the same time, provides IED with a good technique for minimizing the Lagrangian.



---

*Source: 8750740-2018-05-22.xml*