structural optimization under uncertain loads and nodal locations this paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes the second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements we first review the discrete form of the uncertain loads problem which has been previously solved using a weighted average of multiple load patterns with minor modifications we extend this solution to include loads described by continuous joint probability density functions we then proceed to the main contribution of this paper structural optimization under uncertainty in the nodal locations this optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix it is shown however that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads by expressing these equivalent loads in terms of continuous random variables we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper the optimization algorithms are developed in the context of minimum compliance maximum stiffness design simple examples are presented the results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design for structures containing thin substructures under axial loads it is shown that these uncertainties a are of first order significance influencing the linear elastic response quantities and b can affect designs by avoiding unrealistically optimistic and potentially unstable structures the additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem

robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis abstract topology optimization often leads to structures consisting of slender elements which are particularly sensitive to geometric imperfections such imperfections might affect the structural stability and induce large displacement effects in these slender structures this paper therefore presents a robust approach to topology optimization which accounts for geometric imperfections and their potentially detrimental influence on the structural stability geometric nonlinear effects are incorporated in the optimization by means of a total lagrangian finite element formulation in the minimization of end compliance geometric imperfections are modeled as a vector valued random field in the design domain the resulting uncertain performance of the design is taken into account by minimizing a weighted sum of the mean and standard deviation of the compliance in the robust optimization problem these stochastic moments are typically estimated by means of sampling methods such as monte carlo simulation however these methods require multiple independent nonlinear finite element analyses in each design iteration of the optimization algorithm an efficient solution algorithm which uses adjoint differentiation in a second order perturbation method is therefore developed to estimate the stochastic moments during the optimization two applications with structures that exhibit different types of structural instabilities are examined in both cases it is demonstrated by means of an extensive monte carlo simulation that the deterministic design is very sensitive to imperfections while the design obtained by means of the proposed method is much more robust

nonlinear robust optimization via sequential convex bilevel programming in this paper we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min max problems which arise in the context of semi infinite programming here our main motivation are nonlinear inequality constrained robust optimization problems in the first part of the paper we propose a conservative approximation strategy for such nonlinear and non convex robust optimization problems under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given we show how to formulate a lower level concave min max problem which approximates the robust counterpart in a conservative way this approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties in the second part of the paper we review existing theory on optimality conditions for nonlinear lower level concave min max problems which arise in the context of semi infinite programming regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem we end up with a structured mathematical program with complementarity constraints mpcc the special hierarchical structure of this mpcc can be exploited in a novel sequential convex bilevel programming algorithm we discuss the surprisingly strong global and locally quadratic convergence properties of this method which can in this form neither be obtained with existing sqp methods nor with interior point relaxation techniques for general mpccs finally we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example

optimal design of trusses with geometric imperfections accounting for global instability abstract a topology optimization method is proposed for the design of trusses with random geometric imperfections due to fabrication errors this method is a generalization of a previously developed perturbation approach to topology optimization under geometric uncertainties the main novelty in the present paper is that the objective function includes the nonlinear effects of potential buckling due to misaligned structural members solutions are therefore dependent on the magnitude of applied loads and the direction of resulting internal member forces whether they are compression or tension direct differentiation is used in the sensitivity analysis and analytical expressions for the associated derivatives are derived in a form that is computationally efficient a series of examples illustrate how the effects of geometric imperfections and buckling may have substantial influence on truss design monte carlo simulation together with second order elastic analysis is used to verify that solutions offer improved performance in the presence of geometric uncertainties

robust nonlinear optimization with conic representable uncertainty set the robust optimization methodology is known as a popular method dealing with optimization problems with uncertain data and hard constraints this methodology has been applied so far to various convex conic optimization problems where only their inequality constraints are subject to uncertainty in this paper the robust optimization methodology is applied to the general nonlinear programming nlp problem involving both uncertain inequality and equality constraints the uncertainty set is defined by conic representable sets the proposed uncertainty set is general enough to include many uncertainty sets which have been used in literature as special cases the robust counterpart rc of the general nlp problem is approximated under this uncertainty set it is shown that the resulting approximate rc of the general nlp problem is valid in a small neighborhood of the nominal value furthermore a rather general class of programming problems is posed that the robust counterparts of its problems can be derived exactly under the proposed uncertainty set our results show the applicability of robust optimization to a wider area of real applications and theoretical problems with more general uncertainty sets than those considered so far the resulting robust counterparts which are traditional optimization problems make it possible to use existing algorithms of mathematical optimization to solve more complicated and general robust optimization problems

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

a new method for optimal truss topology design truss topology optimization formulated in terms of displacements and bar volumes results in a large nonconvex optimization problem for the case of maximization of stiffness for a prescribed volume this paper presents a new equivalent an unconstrained and convex minimization problem in displacements only where the function to be minimized is the sum of terms each of which is the maximum of two convex quadratic functions existence of solutions is proved as is the convergence of a nonsmooth steepest descent type algorithm for solving the topology optimization problem the algorithm is computationally attractive and has been tested on a large number of examples some of which are presented

potential reduction polynomial time method for truss topology design the multiload truss topology design problem is modeled as a minimization of the maximum with respect to k loading scenarios compliance subject to equilibium constraints and restrictions on the bar volumes the problem may involve a very large number of potential bars so as to allow a rich variety of topologies the original formulation is in terms of two sets of variables m bar volumes and n nodal displacements several equivalent convex reformulations of this problem are presented these convex problems although highly nonlinear possess nice analytical structure and therefore can be solved by an interior point potential reduction method associated with appropriate logarithmic barrier for the feasible domain of the problem for this method to improve the accuracy of the current a proximate solution by an absolute constant factor it suffices in the worst case to perform o sqrt km newton steps with o k 3 n 3 k 2 n 2 m operations per step

robustness analysis of trusses with separable load and structural uncertainties this paper discusses evaluation techniques of the robustness function of trusses which is regarded as one of measures of structural robustness under the uncertainties of member stiffnesses and external forces by using quadratic embedding of the uncertainty and the s procedure we formulate a quasiconvex optimization problem which provides lower bounds of the robustness functions a bisection method is proposed where we solve a finite number of semidefinite programming problems in order to obtain a global optimal solution to the proposed quasiconvex optimization problem the lower bounds of the robustness functions are computed for various trusses under several uncertainty circumstances

robust optimization of sums of piecewise linear functions with application to inventory problems robust optimization is a methodology that has gained a lot of attention in the recent years this is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models unfortunately the second property is usually lost when the cost function that needs to be robustified is not concave or linear with respect to the perturbing parameters in this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set given that these problems are known to be intractable we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed integer linear program and relate this scheme to methods that are based on exploiting affine decision rules our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semidefinite program with the latter having the potential to provide solutions of better quality than the former at the price of heavie

extending scope of robust optimization comprehensive robust counterparts of uncertain problems in this paper we propose a new methodology for handling optimization problems with uncertain data with the usual robust optimization paradigm one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set whereas with the proposed approach we require also a controlled deterioration in performance when the data is outside the uncertainty set the extension of robust optimization methodology developed in this paper opens up new possibilities to solve efficiently multi stage finite horizon uncertain optimization problems in particular to analyze and to synthesize linear controllers for discrete time dynamical systems

on safe tractable approximations of chance constrained linear matrix inequalities in the paper we consider the chance constrained version of an affinely perturbed linear matrix inequality lmi constraint assuming the primitive perturbations to be independent with light tail distributions e g bounded or gaussian constraints of this type playing a central role in chance constrained linear conic quadratic semidefinite programming are typically computationally intractable the goal of this paper is to develop a tractable approximation to these chance constraints our approximation is based on measure concentration results and is given by an explicit system of lmis thus the approximation is computationally tractable moreover it is also safe meaning that a feasible solution of the approximation is feasible for the chance constraint

robust topology optimization under multiple independent unknown but bounded loads abstract the optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable or even lead to catastrophic failures a two level optimization formulation is often used in the robust topology optimization rto under uncertain loads various approaches have been reported to identify the critical loads associated with the worst structure responses because convex model approaches apply convex approximations to the original non convex model at the lower level the optimal designs obtained by these methods are greatly dependent on the quality of the approximation in this paper a new formulation based on the wolfe duality for the rto problems with multiple independent unknown but bounded loads is proposed following the two level formulation the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the wolfe duality both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non convexity associated with the lower level formulation numerical examples are also presented to demonstrate the proposed approach

construction and application of an ellipsoidal convex model using a semi definite programming formulation from measured data abstract as a set theory based convex model the ellipsoidal model provides an attractive framework for treating uncertain but bounded variations in the structural reliability analysis and design optimization however improper modeling of the uncertainties may give rise to misleading non probabilistic reliability analysis thus result in either unsafe or over conservative designs this paper presents a systematic study on the mathematical formulation for constructing the minimum volume ellipsoidal convex model using a given set of sample data and shows its application in existing methods of non probabilistic reliability analysis and design optimization of structures with bounded uncertainties in this method the uncertain parameters are first divided into groups according to their sources for each individual group of uncertainties the minimum volume ellipsoid problem is reformulated into a semi definite programming sdp problem and thus can be efficiently solved to its global optimum further a linear transformation based on the eigenvalue analysis is employed to map the ellipsoidal model into a standard uncertainty space this uncertainty modeling technique enables a compact and differentiable bound description of the parameter variations moreover it has another useful property the affine invariance which is shown to be necessary for meaningful definition of a non probabilistic reliability index the effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources

robust solutions of uncertain quadratic and conic quadratic problems we consider a conic quadratic and in particular a quadratically constrained optimization problem with uncertain data known only to reside in some uncertainty set cal u the robust counterpart of such a problem leads usually to an np hard semidefinite problem this is the case for example when cal u is given as the intersection of ellipsoids or as an n dimensional box for these cases we build a single explicit semidefinite program which approximates the np hard robust counterpart and we derive an estimate on the quality of the approximation which is essentially independent of the dimensions of the underlying conic quadratic problem

design of planar articulated mechanisms using branch and bound this paper considers an optimization model and a solution method for the design of two dimensional mechanical mechanisms the mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously the underlying mechanical analysis model is based on a truss representation allowing for large displacements for mechanisms undergoing large displacements elastic stability is of major concern we derive conditions modeled by nonlinear matrix inequalities which guarantee that a stable equilibrium is found and that buckling is prevented the feasible set of the design problem is described by nonlinear differentiable and non differentiable constraints as well as nonlinear matrix inequalities to solve the mechanism design problem a branch and bound method based on convex relaxations is developed to guarantee convergence of the method two different types of convex relaxations are derived the relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub problems encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality

robust topology optimization minimization of expected and variance of compliance robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies this paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude using exact formulations and analytically derived sensitivities it shows that only a few additional load cases are required which scales in polynomial time with the number of uncertain load cases the approach is implemented using the level set topology optimization method shape sensitivities are derived using the adjoint method several examples are used to investigate the effect of including variance in robust compliance optimization

stochastic control with uncertain parameters via chance constrained control chance constrained methods handle problem uncertainty by formulating the constraints probabilistically and only guaranteeing their satisfaction up to a given violation level this paper extends existing chance constrained methods to handle both uncertainty in the system state as well as in the constraint parameters due to the imperfect knowledge of the system state caused by motion sensor and environment uncertainty the system constraints cannot be guaranteed to be satisfied and consequently must be considered probabilistically prior work has primarily only dealt with uncertainty in the system state or constraint parameters but not both simultaneously however in many motivating applications both frequently occur to handle this case previous approaches either approximate the distribution leading to a nonconvex optimization program or use sampling alone to represent the uncertainty which requires a large number of samples to accurately represent the distribution to address these limitations a novel hybrid method is proposed that uses both analytical functions and sampling to represent the uncertainty it is shown that under certain conditions the resulting optimization program using this hybrid representation is convex to check the convexity an efficient a priori sufficient condition is developed furthermore by using this hybrid representation this method drastically reduces the computational complexity over previous methods which is demonstrated through two examples consequently this method has the ability to enable real time stochastic control for the motivating applications

on normal and subnormal q ary codes the authors extend to the q ary case the notions of a normal code a subnormal code and the amalgamated direct sum construction in order to investigate problems related to the covering radius of codes for example the authors prove that every nonbinary nontrivial perfect code is absubnormal they also include some linear programming lower bounds on ternary codes with covering radius 2 or 3