Solving ill-posed bilevel programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem

Variational Principles for Set-Valued Mappings with Applications to Multiobjective Optimization

This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which~being certainly of independent interest are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang formulated via an appropriate subdifferential notion for set-valued mappings with values in partially ordered spaces. Based on these variational principles and corresponding tools of generalized differentiation, we derive new conditions of the coercivity and Palais-Smale types ensuring the existence of optimal solutions to set-valued optimization problems with noncompact feasible sets in infinite dimensions and then obtain necessary optimality and suboptimality conditions for nonsmooth multiobjective optmization problems with general constraints, which are new in both finite-dimensional and infinite-dimensional settings

Stability and Error Analysis for Optimization and Generalized Equations

Stability and error analysis remain challenging for problems that lack regularity properties near solutions, are subject to large perturbations, and might be infinite dimensional. We consider nonconvex optimization and generalized equations defined on metric spaces and develop bounds on solution errors using the truncated Hausdorff distance applied to graphs and epigraphs of the underlying set-valued mappings and functions. In the process, we extend the calculus of such distances to cover compositions and other constructions that arise in nonconvex problems. The results are applied to constrained problems with feasible sets that might have empty interiors, solution of KKT systems, and optimality conditions for difference-of-convex functions and composite functions

Characterizations of solutions for vector equilibrium problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems

Exploiting linkage information in real-valued optimization with the real-valued gene-pool optimal mixing evolutionary algorithm

The recently introduced Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) has been shown to be among the state-of-the-art for solving discrete optimization problems. Key to the success of GOMEA is its ability to efficiently exploit the linkage structure of a problem. Here, we introduce the Real-Valued GOMEA (RV-GOMEA), which incorporates several aspects of the real-valued EDA known as AMaLGaM into GOMEA in order to make GOMEA well-suited for real-valued optimization. The key strength of GOMEA to competently exploit linkage structure is effectively preserved in RV-GOMEA, enabling excellent performance on problems that exhibit a linkage structure that is to some degree decomposable. Moreover, the main variation operator of GOMEA enables substantial improvements in performance if the problem allows for partial evaluations, which may be very well possible in many real-world applications. Comparisons of performance with state-of-the-art algorithms such as CMA-ES and AMaLGaM on a set of well-known benchmark problems show that RV-GOMEA achieves comparable, excellent scalability in case of black-box optimization. Moreover, RV-GOMEA achieves unprecedented scalability on problems that allow for partial evaluations, reaching near-optimal solutions for problems with up to millions of real-valued variables within one hour on a normal desktop computer

A steepest descent method for set optimization problems with set-valued mappings of finite cardinality

In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020)

Suboptimality Conditions for Mathematical Programs with Equilibrium Constraints

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techriiques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle

Global algorithms for nonlinear discrete optimization and discrete-valued optimal control problems

Optimal control problems arise in many applications, such as in economics, finance, process engineering, and robotics. Some optimal control problems involve a control which takes values from a discrete set. These problems are known as discrete-valued optimal control problems. Most practical discrete-valued optimal control problems have multiple local minima and thus require global optimization methods to generate practically useful solutions. Due to the high complexity of these problems, metaheuristic based global optimization techniques are usually required.One of the more recent global optimization tools in the area of discrete optimization is known as the discrete filled function method. The basic idea of the discrete filled function method is as follows. We choose an initial point and then perform a local search to find an initial local minimizer. Then, we construct an auxiliary function, called a discrete filled function, at this local minimizer. By minimizing the filled function, either an improved local minimizer is found or one of the vertices of the constraint set is reached. Otherwise, the parameters of the filled function are adjusted. This process is repeated until no better local minimizer of the corresponding filled function is found. The final local minimizer is then taken as an approximation of the global minimizer.While the main aim of this thesis is to present a new computational methodfor solving discrete-valued optimal control problems, the initial focus is on solvingpurely discrete optimization problems. We identify several discrete filled functionstechniques in the literature and perform a critical review including comprehensive numerical tests. Once the best filled function method is identified, we propose and test several variations of the method with numerical examples.We then consider the task of determining near globally optimal solutions of discrete-valued optimal control problems. The main difficulty in solving the discrete-valued optimal control problems is that the control restraint set is discrete and hence not convex. Conventional computational optimal control techniques are designed for problems in which the control takes values in a connected set, such as an interval, and thus they cannot solve the problem directly. Furthermore, variable switching times are known to cause problems in the implementation of any numerical algorithm due to the variable location of discontinuities in the dynamics. Therefore, such problem cannot be solved using conventional computational approaches. We propose a time scaling transformation to overcome this difficulty, where a new discrete variable representing the switching sequence and a new variable controlling the switching times are introduced. The transformation results in an equivalent mixed discrete optimization problem. The transformed problemis then decomposed into a bi-level optimization problem, which is solved using a combination of an efficient discrete filled function method identified earlier and a computational optimal control technique based on the concept of control parameterization.To demonstrate the applicability of the proposed method, we solve two complex applied engineering problems involving a hybrid power system and a sensor scheduling task, respectively. Computational results indicate that this method is robust, reliable, and efficient. It can successfully identify a near-global solution for these complex applied optimization problems, despite the demonstrated presence of multiple local optima. In addition, we also compare the results obtained with other methods in the literature. Numerical results confirm that the proposed method yields significant improvements over those obtained by other methods