# Engineering Design by Geometric Programming

Authors: Chia-Hui Huang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions, where all functions are of signomial form. The importance of GP comes from two relatively recent developments: (i) new methods can solve even large-scale GP extremely efficiently and reliably; (ii) a number of practical problems have recently been found to be equivalent to or approximated by GP. This study proposes an optimization approach for solving GP. Our approach is first to convert all signomial terms in GP into convex and concave terms. Then the concave terms are further treated with the proposed piecewise linearization method where only⌈log2(m-1)⌉ binary variables are used. It has the following features: (i) it offers more convenient and efficient means of expressing a piecewise linear function; (ii) fewer 0-1 variables are used; (iii) the computational results show that the proposed method is much more efficient and faster than the conventional one, especially when the number of break points becomes large. In addition, the engineering design problems are illustrated to evaluate the usefulness of the proposed methods.


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## Body

## 1. Introduction

Consider the following geometric program (GP) [1]:
(1)minf0(x)(2)s.t.fk(x)≤1,k=1,2,…,p,(3)gl(x)=1,l=1,2,…,q,(4)x=(x1,x2,…,xn),
where gl(x), l=1,2,…,q, are monomials as in (5), cl≥0, and the exponential constants αil∈ℝ, i=1,2,…,n, l=1,2,…,q:
(5)gl(x)=cl∏i=1n‍xiαil
and fk(x), k=0,1,…,p, are posynomials as in (6) which are sum of one or more monomials, chk≥0, and αihk∈ℝ,i=1,2,…,n, h=1,2,…,Hk,k=0,1,…,p:
(6)fk(x)=∑h=1Hk‍chk∏i=1n‍xiαihk.GP has been applied in many fields of applications including analog/digital circuit design [2–8], chemical engineering [9–11], mechanical engineering [12–19], power control  [20], and communication network systems [21–23].Obtaining the optimal solutions for GP is not straightforward because the signomial terms in the objective function and constraints cannot be solved directly. As a result, many approaches have been developed. Coello and Cortés [24] proposed a genetic algorithm with an artificial immune system to solve a GP in engineering optimization. Nevertheless, this method can only obtain the local optima. Horst and Tuy [25] introduced an analytical approach for solving a problem with Lipschitzian objective and constraints. The restriction of this approach is to find the global optimum only if the range of variables can be reduced by analytical techniques. Sherali and Tuncbilek [26] developed a reformulation-linearization technique (RLT) which generates polynomial implied constraints and then linearizes the resulting problem by introducing new variables. Lin and Tsai [14] introduced a generalized method to find multiple optimal solutions of signomial discrete programming problems with free variables. By means of variable substitution and convexification strategies, a signomial discrete programming problem with free variables is first converted into another convex mixed-integer nonlinear programming problem solvable to obtain an exactly global optimum. Tsai [17] proposed a novel method to solve signomial discrete programming problems. The signomial terms are first convexified following the convexification strategies. The original program is then converted into a convex integer program solvable by commercialized packages to obtain globally optimal solutions. Tsai and Lin [19] developed an efficient method to solve a posynomial geometric program with separable functions. Power transformations and exponential transformations are utilized to convexify and underestimate posynomial terms. The original program therefore can be converted into a convex mixed-integer nonlinear program solvable to obtain a global optimum.This paper develops an optimization approach for solving GP. Our approach is listed as follows.(i)
Convert all signomial terms into convex and concave ones.(ii)
The concave terms are further treated with the proposed piecewise linearization method where only⌈log2(m-1)⌉ binary variables are used.The rest of this paper is organized as follows. Section2 introduces the proposed methods. Section 3 provides some numerical examples to illustrate the modeling idea and the usefulness of the proposed method. Section 4 gives our conclusions.


## 2. Proposed Methods

In this section, the proposed methods will be presented. First, we recall what a signomial function is. A functionf:ℝ++n→ℝ defined as
(7)f(x)=c∏i=1n‍xiαi,
where c>0 and αi∈ℝ for all i, is called a monomial function or simply a monomial [1]. Note that the exponents αi of a monomial can be any real numbers, but the coefficient c must be nonnegative. A sum of monomials, namely, a function of the form
(8)f(x)=∑k=1N‍ck∏i=1n‍xiαik,
where ck>0 and αik∈ℝ, is called a posynomial function with N terms or simply a posynomial. A signomial is a linear combination of monomials of some positive variables x1,…,xn.Letf be defined on an open convex set D⊆ℝn and be twice differentiable; it is known that (i) f is convex on D if and only if the Hessian matrix ∇2f(x) is positive semidefinite (p.s.d. for short) at each x∈D; (ii) if ∇2f(x) is positive definite (p.d. for short) at each x∈D, then f is strictly convex. For convenience, we denote by Δk the leading principal minors of A.Lemma 1.
LetA be a nonzero n×n symmetric matrix.(a)
IfA is positive semidefinite, then Δk≥0,k=1,2,…,n, and not all Δk=0.(b)
A is positive definite if and only if Δk>0, for all k=1,2,…,n.The converse of Lemma1(a) is false. For example, let A=[10000000-1]; we have 〈x,Ax〉=x12-x32 which is not always nonnegative for all x∈ℝ3. But Δ1=1≥0, Δ2=0≥0, and Δ3=0≥0. In fact, the converse of Lemma 1(a) is true only for n=2; see [27, page 112].Lemma 2.
LetA be a nonzero symmetric matrix. Then, the following will hold.(a)
A is p.s.d. if and only if all of its eigenvalues are nonnegative.(b)
A is p.d. if and only if all of its eigenvalues are positive.Proposition 3.
Letf1:ℝ++n→ℝ be defined as f1(x)=c1∏i=1n‍xiαi, where c1>0 and αi≤0 for all i=1,2,…,n. Then f1 is a convex function.Proof.
Sincec1>0, it is enough to show that f~1(x)=∏i=1n‍xiαi is convex. Let g(x)=lnf~1(x)=∑i=1n‍lnxiαi=∑i=1n‍αilnxi. Then, we have
(9)∇g(x)=[α1x1α2x2⋯αnxn]T,∇2g(x)=[-α1x120⋯00-α2x22⋯0⋮⋮⋱⋮00⋯-αnxn2].
Because αi≤0 for all i=1,2,…,n, we know that all eigenvalues of ∇2g(x) are nonnegative, which implies (by Lemma 1(a)) that ∇2g(x) is positive semidefinite. Thus, g(x)=lnf(x) is a convex function which yields that f~1(x) is a convex function.Proposition 4.
Letf2:ℝ++n→ℝ be defined as f2(x)=c2∏i=1n‍xiαi, where c2<0 and αi>0 for all i=1,2,…,n with 1-∑i=1n‍αi≥0. Then f2 is a convex function.Proof.
It is not hard to compute that[∇f2(x)]i=c2αixiαi-1∏j=1,j≠in‍xjαj. In other words,
(10)∇f2(x)=[c2α1x1α1-1x2α2⋯xnαnc2α2x1α1x2α2-1⋯xnαn⋮c2αnx1α1x2α2⋯xnαn-1].
In addition, it can be verified that
(11)[∇2f2(x)]ij=∂2f2(x)∂xi∂xj={αiαj∂xi∂xjf2(x),i≠j,αi(αi-1)∂xi2f2(x),i=j,
Namely,(12)∇2f2(x)=[c2α1(α1-1)x1-2∏i=1n‍xiαic2α1α2x1-1x2-1∏i=1n‍xiαi⋯c2α1αnx1-1xn-1∏i=1n‍xiαic2α2α1x2-1x1-1∏i=1n‍xiαic2α2(α2-1)x2-2∏i=1n‍xiαi⋯c2α2αnx2-1xn-1∏i=1n‍xiαi⋮⋮⋱⋮c2αnα1xn-1x1-1∏i=1n‍xiαic2αnα2xn-1x2-1∏i=1n‍xiαi⋯c2αn(αn-1)xn-2∏i=1n‍xiαi].
Moreover, the determinant of∇2f2(x) can be computed and be shown by induction as
(13)det[∇2f2(x)]=(-c2)n(∏i=1n‍αixinαi-2)(1-∑i=1n‍αi).
We will complete the proof by discussing the following two cases.
Case i. If 1-∑i=1n‍αi=0, then it is not hard to verify that ∇2f2(x)x is a zero vector for any x∈ℝn. Hence, xT∇2f2(x)x=0 which says that ∇2f2(x) is a positive semidefinite matrix by definition. Therefore, f2(x) is convex under this case.
Case ii. If 1-∑i=1n‍αi>0, then we know from (13) that
(14)Δi=(-c2)i(∏j=1i‍αjxjiαj-2)(1-∑j=1i‍αj),
where Δi denotes the ith principal minor of the Hessian matrix of f2(x). Note that c2<0, αi>0 for all i=1,2,…,n, and 1-∑i=1n‍αi>0. Therefore, it can be seen that Δi>0 for all i=1,2,…,n, which implies (by Lemma 1(b)) that ∇2f2(x) is a positive definite matrix. This says that f2(x) is strictly convex under this case.
From all the previous, the desired result follows.We want to point out that our results also provide an alternative proof for the main result(Theorem7) of [28]. Indeed, Maranas and Floudas [28] further discuss another condition as follows:
(15)∃jsuchthatαj≥1-∑i≠jn‍αi,αi≤0,∀i≠j,i=1,2,…n,
to guarantee that f1 defined as in Proposition 3 is a convex function. Our approach can be also employed to verify this fact. To see this, we arrange all powers αi in a decreasing order. In other words, without loss of generality, we assume that
(16)α1>α2≥⋯≥αn.
Notice that condition (15) implies that α1 is positive and all the other α2,…,αn are nonpositive with α1≥1-∑i=2n‍αi. As mentioned in Proposition 3, we only need to show that the function f~1(x)=∏i=1n‍xiαi is convex. By similar arguments as in the proof of Proposition 4, we know that
(17)Δ^i=(-1)i(∏j=1i‍αjxjiαj-2)(1-∑j=1i‍αj),
where Δ^i denotes the ith principal minor of the Hessian matrix of f~1(x). From conditions (15) and (16), it is easily verified that (1-∑j=1i‍αj)<0 for each i. It is also not hard to observe that ∏j=1i‍αj is positive if i is odd and is negative if i is even. In summary, there holds
(18)(-1)i(∏j=1i‍αjxjiαj-2)<0.
The above two inequalities yield that Δ^i>0 for each i. Thus, following the same arguments as in Proposition 4, we can conclude that f1 is also a convex function under condition (15).Remark 5.
Ifc>0, αi<0, for i=1,2,…,m, and αi>0, i=m+1,m+2,…,n, then f(x) can be converted into the following convex function:
(19)f′(x)=c(∏i=1m‍xiαi)(∏i=m+1n‍zi-αi),
where zi=xi-1, i=m+1,m+2,…,n.Remark 6.
Ifc<0, αi>0, for i=1,2,…,m, and αi<0, i=m+1,m+2,…,n, then f(x) can be converted into the following convex function:
(20)f′(x)=c(∏i=1m‍ziαi/t)(∏i=m+1n‍zi-αi/t),
where (i)
t is the smallest positive integer satisfying 1-(1/t)(∑i=1m‍αi-∑i=m+1n‍αi)≥0.(ii)
zi=xit, for αi>0, i=1,2,…,m.(iii)
zi=xi-t, for αi<0, i=m+1,m+2,…,n.Obviously, all signomial terms are converted into convex and concave terms by the proposed methods. All concave terms in the GP problem can be further represented by an effective piecewise linearization method as follows.Conventional methods for linearizing a concave function withm break points require m-1 binary variables. However, when m becomes large, the computation will be very time consuming and may cause a heavy computational burden. An effective piecewise linearization method proposed by Huang [29] is presented in which only ⌈log2(m-1)⌉ binary variables are used.Consider a grid ofm points throughout the interval of [a1,am] and denote the break points a1,a2,…,am, where a1<a2<⋯<am. f(ak) is evaluated at each ak, and two adjacent break points (ak,f(ak)) and (ak+1,f(ak+1)) are connected with a straight line as shown in Figure 1.Figure 1
A piecewise linear function off(x).Letθ be an integer, 1≤θ≤m-1, expressed as
(21)θ=1+∑j=1h‍2j-1uj,
where h=⌈log2(m-1)⌉ and uj∈{0,1}. Given the values of uj, 1≤j≤h, there is a unique θ.LetG(θ)⊆{1,2,…,h} be a set composed of all indices corresponding to a given value θ, such that
(22)∑j∈G(θ)2j-1=θ-1.For instance,G(1)=⌀, G(2)={1}, G(3)={2}, G(4)={1,2}, …, G(10)={1,4}, and so forth.Denote by∥G(θ)∥ the number of elements in G(θ). For instance, given 1≤x≤11, then h=⌈log2(m-1)⌉=4. ∥G(1)∥=0, ∥G(2)∥=1, ∥G(4)∥=2, and so forth. A list of all θ, uj, G(θ), and ∥G(θ)∥ is shown in Table 1.Table 1
Information of interval ofx.
Interval ofx
θ
u
1
u
2
u
3
u
4
G
(
θ
)
∥
G
(
θ
)
∥
1
≤
x
≤
2
1
0
0
0
0
⌀
0
2
≤
x
≤
3
2
1
0
0
0
1
1
3
≤
x
≤
4
3
0
1
0
0
2
1
4
≤
x
≤
5
4
1
1
0
0
1, 2
2
5
≤
x
≤
6
5
0
0
1
0
3
1
6
≤
x
≤
7
6
1
0
1
0
1, 3
2
7
≤
x
≤
8
7
0
1
1
0
2, 3
2
8
≤
x
≤
9
8
1
1
1
0
1, 2, 3
3
9
≤
x
≤
10
9
0
0
0
1
4
1
10
≤
x
≤
11
10
1
0
0
1
1, 4
2Now we intend to usek as an index to indicate that the kth interval is activated, which results in ak≤x≤ak+1 and f^(x)=f(ak)+sk(x-ak).Definem-1 functions qθ based on the combination of uj as follows:
(23)qθ=∥G(θ)∥-∑uj∈D,j∈G(θ)‍uj+∑uj∈D,j∉G(θ)‍uj,
where θ=1,2,…,m-1, j=1,2,…,h, and h=⌈log2(m-1)⌉.The following proposition is deduced.Proposition 7.
Given a set of binary variablesD={u1,u2,…,uh}, where uj∈{0,1}, j=1,2,…,h, and h=⌈log2(m-1)⌉, there is one and only one k,1≤k≤m-1, such that (i)
qk=0and∥G(k)∥=∑uj∈D,j∈G(k)‍uj-∑uj∈D,j∉G(k)‍uj,(ii)
qθ≥1for1≤θ≤m-1,θ≠k.Taking Table1 for instance, given 1≤x≤11, m=11. Let D={u1,u2,u3,u4}={1,0,0,1}; then we have
(24)q1=∥G(1)∥+u1+u2+u3+u4=0+1+0+0+1=2q2=∥G(2)∥-u1+u2+u3+u4=1-1+0+0+1=1⋮q8=∥G(8)∥-u1-u2-u3+u4=3-1-0-0+1=3q9=∥G(9)∥+u1+u2+u3-u4=1+1+0+0-1=1q10=∥G(10)∥-u1+u2+u3-u4=2-1+0+0-1=0.Sinceq10=0 and qθ≥1 for 1≤θ≤10, θ≠10, then only a10≤x≤a11 is activated and results in f^(x)=f(a10)+s10(x-a10).We then have the following theorem.Theorem 8.
Consider a piecewise linear functionf^(x) of a concave function f(x), where x is divided into m-1 intervals by m break points, a1<a2<⋯<am. Given a set of binary variables D={u1,u2,…,uh}, where h=⌈log2(m-1)⌉,x and f^(x) can be expressed as the following linear inequalities:
(25)aθ-Mqθ≤x≤aθ+1+Mqθ,(26)f(aθ)+sθ(x-aθ)-M′qθ≤f^(x)≤f(aθ)+sθ(x-aθ)+M′qθ,(27)qθ=∥G(θ)∥-∑uj∈D,j∈G(θ)‍uj+∑uj∈D,j∉G(θ)‍uj,
where sθ=(f(aθ+1)-f(aθ))/(aθ+1-aθ),θ=1,2,…,m-1. M, and M′ are large positive values,M=max{am,1} and M′=max{f(a1),f(a2),…,f(am),1}.Proof.
Given the value of a specificD=D0, D0={u10,u20,…,uh0}, consider the following two cases.
Case i. There is one and only one integer θ0, θ0=1+∑j=1h‍2j-1uj0 such that (a)
G(θ0)={j∣uj0=1,(b)
∥G(θ0)∥=∑j∈G(θ0)‍uj0,(c)
∑uj0∈D,j∉G(θ0)‍uj0=0. Expressions (a), (b), and (c) force qθ0=0 (from (27)). Then, aθ0≤x≤aθ0+1 is activated (from (25)) and results in f^(x)=f(aθ0)+sθ0(x-aθ0) (from (26)).
Case ii. For other m-1 integers 1≤θ′≤m, θ′≠θ0, since ∥G(θ′)∥>∑j∈G(θ′)‍uj0+∑uj0∈D,j∉G(θ′)‍uj0, we then have qθ′≥1. In this case, inequalities (25) and (26) are still correct but not being activated.Compared with the conventional piecewise linearization methods [30–36], the number of newly added binary variables in the proposed method for a piecewise linear function with m break points is significantly reduced from m-1 to ⌈log2(m-1)⌉. The algorithm is listed in Algorithm 1 based on Theorem 8.Algorithm 1 (the proposed piecewise linearization method).
Step
1. Identify the number of subintervals and choose the break points, ai, i=1,2,…,m.
Step
2. Calculate the values of  h=⌈log2m-1⌉ based on the number of break points selected and obtain qθ, θ=1,2,…,m-1, based on the combination of uh as (27).
Step
3. Generate the set of constraints for x and f^(x) as (25) and (26), respectively, and solve the resulting problem.


## 3. Numerical Examples

In this section, we have conducted some engineering design problems to evaluate the usefulness of the proposed methods.Example 1.
Consider the following engineering design problem of a speed reducer which was proposed by Golinski [37] as in Figure 2.Figure 2
Schematic of the welded beam problem, Example2.The objective of this problem is to minimize the weight of the speed reducer while satisfying a number of constraints imposed by gear and shaft design practices. There are seven design variables,x1 (width of the gear face, cm), x2 (teeth module, cm), x3 (number of pinion teeth), x4 (shaft 1 length between bearings, cm), x5 (shaft 2 length between bearings, cm), x6 (diameter of shaft 1, cm), and x7 (diameter of shaft 2, cm). The constraints are characterized by h1 (upper bound on the bending stress of the gear tooth), h2 (upper bound on the contact stress of the gear tooth), h3, h4 (upper bounds on the transverse deflection of shafts 1, 2), h5, h6 (upper bounds on the stresses in shafts 1, 2), h7, h8, h9 (dimensional restrictions based on space and experience), and h10, h11 (dimensional requirements for shafts based on experience).The problem can be formulated as(28)min2.6179x1x22x32+11.7287x1x22x3-33.84556x1x22-1.508x1x62-1.508x1x72+7.4777x63+7.4777x73+0.7854x4x62+0.7854x5x72(29)s.t.h1=-x1x22x3+27.0≤0,(30)h2=-x1x22x32+397.5≤0,(31)h3=-x2x3x64+1.93x43≤0,(32)h4=-x2x3x74+1.93x53≤0,(33)h5=555025.0x42+1.690×107x22x32-12100.0x22x32x66≤0,(34)h6=555025.0x52+1.575×108x22x32-7225.0x22x32x76≤0,(35)h7=x2x3-40.0≤0,(36)h8=-x1+5x2≤0,(37)h9=x1-12x2≤0,(38)h10=-x4+1.5x6+1.9≤0,(39)h11=-x5+1.1x7+1.9≤0,(40)2.6≤x1≤3.6,0.7≤x2≤0.8,17.0≤x3≤28.0,7.3≤x4≤8.3,(41)7.3≤x5≤8.3,2.9≤x6≤3.9,5.0≤x7≤5.5.This problem has been solved by Coello and Cortés [24] using genetic algorithm. The solution is (x1,x2,x3,x4,x5,x6,x7) = (3.5, 0.7, 17.0, 7.300008, 7.715322, 3.350215, 5.286655) with the objective value 2994.3419.For instance, the nonlinear termsx1x22x32, x1x22x3, and x4x62 can be convexified as z1,1-1z2,1-2z3,1-2, z1,1-1z2,1-2z3,1-1, and z4,1-1z6,1-2 by Remark 5, where zk,1=xk-1, k=1,2,…,7. The nonlinear terms -x1x22, -x1x62, and -x1x72 can be convexified as -z1,21/3z2,22/3, -z1,21/3z6,22/3, and -z1,21/3z7,22/3 by Remark 6, where z1,2=x13, z2,2=x23, z6,2=x63, and z7,2=x73. The concave terms can be further represented by the proposed piecewise linearization method. Take z1,1=x1-1 for instance; suppose 41 equal-distance break points are used to linearize z1,1=x1-1, as given by a1,1=2.600,a1,2=2.625,a1,3=2.650,…,anda1,41=3.600. Then z1,1=x1-1 can be represented by Algorithm 2 as follows.Algorithm 2.
Step
1. The proposed method requires the use of ⌈log2(41-1)⌉=6 binary variables to linearize z1,1=x1-1 in a piecewise manner. Compared with the conventional piecewise linearization methods, the number of newly added binary variables for a piecewise linear function with 41 break points is significantly reduced from 40 to 6.
Step
2.
h
1,1
=
⌈
log
2
(
41
-
1
)
⌉
=
6. A set of q1,θ, θ=1,2,…,40, based on the combination of u1,h1,1 is expressed as (42)(42)q1,1=0+u1,1+u1,2+u1,3+u1,4+u1,5+u1,6q1,2=1-u1,1+u1,2+u1,3+u1,4+u1,5+u1,6⋮q1,40=4-u1,1-u1,2-u1,3+u1,4+u1,5-u1,6.
Step
3. Based on Theorem 8, x1 and z1,1 can be expressed as the following linear inequalities:(43)2.600-Mq1,1≤x1≤2.625+Mq1,12.625-Mq1,2≤x1≤2.650+Mq1,2⋮3.575-Mq1,40≤x1≤3.600+Mq1,40,(44)2.600-1+s1,1(x1-2.600)-M′q1,1≤z1,1≤2.600-1+s1,1(x1-2.600)+M′q1,12.625-1+s1,2(x1-2.625)-M′q1,2≤z1,1≤2.625-1+s1,2(x1-2.625)+M′q1,2⋮3.575-1+s1,40(x1-3.575)-M′q1,40≤z1,1≤3.575-1+s1,40(x1-3.575)+M′q1,40,
where s1,θ=(a1,θ+1-1-a1,θ-1)/(a1,θ+1-a1,θ), θ=1,2,…,40. M and M′ are large positive values.Solving the transformed program with MATLAB R13, Table2 shows the results computed by Coello and Cortés [24] and the proposed method.Table 2
Comparison of Example1.
Item
Coello and Cortés [24]
Proposed method
x
1
3.50
3.4956520
x
2
0.70
0.7000002
x
3
17.0
17.0
x
4
7.300008
7.30000007
x
5
7.715322
7.7120386
x
6
3.350215
3.3433720
x
7
5.286655
5.2853520
Objective
2994.3419
2989.5140Example 2 (see [38]).
Consider the design of a welded beam which is to be rigidly attached to a fixed support and is subjected to a load as illustrated in Figure3. The welded beam is to consist of 1010 steel and is to support a force F of 6000 lb. The length L is assumed to be specified at 14 in.Figure 3
Golinski’s speed reducer problem, Example1.Three methods (brazed, bolted, and welded connections) of attachment and three materials (steel, brass, and aluminum) are considered. The choices of attachment and materials are to be made, which minimizes cost while satisfying a limit on the allowable bending stress, the buckling load, and the allowable deflection at the load application point. The various materials have different elastic, shear moduli, and different costs.The design variables are defined asx1 (a continuous variable for thickness of weld), x2 (a continuous variable for width of beam), x3 (a continuous variable for length of connection), x4 (a continuous variable for thickness of beam), x5 (a continuous variable for diameter of bolts), x6 (a 0-1 variable for brazed connections), x7 (a 0-1 variable for bolted connections), x8 (a 0-1 variable for welded connections), x9 (a 0-1 variable for steel material), x10 (a 0-1 variable for brass material), and x11 (a 0-1 variable for aluminum material).The problem can be formulated as (45)–(50)
(45)min1.2x2x3x6+x5x7+x12x3x8+0.04811Lx2x4x9+0.04811x2x3x4x9+0.06Lx2x4x10+0.06x2x3x4x10+0.015Lx2x4x11+0.015x2x3x4x11(46)s.th1=-σallowable+6FLx22x4≤0,(47)h2=-Pc+6000≤0,(48)h3=-δmax+4FL3Ex23x4≤0,(49)x6+x7+x8=1,(50)x9+x10+x11=1,
where σallowable is the maximum bar bending stress. Pc is the bar buckling load which is a function of the beam dimensions and material properties. δmax is the maximum bar deflection. E is the Young’s modulus, 30×106 psi. More detailed description including the derivation of Pc can be found in Reklaitis et al. [38].Following the same procedure as in Example1, the nonlinear terms x2x3x6, x5x7, x12x3x8, x2x4x9, x2x3x4x9, x2x4x10, x2x3x4x10, x2x4x11, and x2x3x4x11 can be convexified by Remark 5. The concave terms are further represented by the proposed piecewise linearization method. Solving the transformed program with MATLAB R13, we can obtain the global optimum with (x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11)=(0.320,14.109,5.720,0.321,4.992,0,0,1,0,0,1) and the objective value is 1.969 within the tolerance error 0.0001.


## 4. Conclusions

This paper proposes an optimization approach for solving geometric programming problems. Our approach is first to convert all signomial terms in GP into convex and concave terms by the proposed methods. Then the concave terms are further treated with the proposed piecewise linearization method where only⌈log2(m-1)⌉ binary variables are used. It has the following features.(i)
It offers more convenient and efficient means of expressing a piecewise linear function.(ii)
Fewer0-1 variables are used.(iii)
The computational results show that the proposed method is much more efficient and faster than the conventional one, especially when the number of break points becomes large.Numerical examples in real applications are illustrated to demonstrate the usefulness of the proposed methods.



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*Source: 568098-2013-08-22.xml*