Some flows in shape optimization

Geometric flows related to shape optimization problems of Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele-Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain existence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed: we prove that the solutions converge to a generalized Bernoulli exterior free boundary problem

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although their convergence to the Pareto optimal front (PF) is in several instances superior to that of Pareto-based algorithms, the problem of selecting a way to distribute or guide these solutions in a high-dimensional space has not been explored. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying evolutionary algorithm toward specific regions of interest or the entire Pareto front with the desired distribution of Pareto optimal solutions. Additionally, it is shown that generalized decomposition simplifies many-objective problems by unifying the three performance objectives of multi-objective evolutionary algorithms – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method (CE), is created to illustrate the benefits of the proposed concept for many objective problems. This choice of EDA also enables the test of the hypothesis that low-order statistics based EDAs can have comparable performance to more elaborate EDAs

Optimization and Equilibrium Problems with Equilibrium Constraints

The paper concerns optimization and equilibrium problems with the so-called equilibrium constraints (MPEC and EPEC), which frequently appear in applications to operations research. These classes of problems can be naturally unified in the framework of multiobjective optimization with constraints governed by parametric variational systems (generalized equations, variational inequalities, complementarity problems, etc.). We focus on necessary conditions for optimal solutions to MPECs and EPECs under general assumptions in finite-dimensional spaces. Since such problems are intrinsically nonsmooth, we use advanced tools of generalized differentiation to study optimal solutions by methods of modern variational analysis. The general results obtained are concretized for special classes of MPECs and EPECs important in applications

Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms

Tree projections provide a unifying framework to deal with most structural decomposition methods of constraint satisfaction problems (CSPs). Within this framework, a CSP instance is decomposed into a number of sub-problems, called views, whose solutions are either already available or can be computed efficiently. The goal is to arrange portions of these views in a tree-like structure, called tree projection, which determines an efficiently solvable CSP instance equivalent to the original one. Deciding whether a tree projection exists is NP-hard. Solution methods have therefore been proposed in the literature that do not require a tree projection to be given, and that either correctly decide whether the given CSP instance is satisfiable, or return that a tree projection actually does not exist. These approaches had not been generalized so far on CSP extensions for optimization problems, where the goal is to compute a solution of maximum value/minimum cost. The paper fills the gap, by exhibiting a fixed-parameter polynomial-time algorithm that either disproves the existence of tree projections or computes an optimal solution, with the parameter being the size of the expression of the objective function to be optimized over all possible solutions (and not the size of the whole constraint formula, used in related works). Tractability results are also established for the problem of returning the best K solutions. Finally, parallel algorithms for such optimization problems are proposed and analyzed. Given that the classes of acyclic hypergraphs, hypergraphs of bounded treewidth, and hypergraphs of bounded generalized hypertree width are all covered as special cases of the tree projection framework, the results in this paper directly apply to these classes. These classes are extensively considered in the CSP setting, as well as in conjunctive database query evaluation and optimization

Unveiling Hidden Values of Optimization Models with Metaheuristic Approach

Considering that the decision making process for constrained optimization problem is based on modeling, there is always room for alternative solutions because there is usually a gap between the model and the real problem it depicts. This study looks into the problem of finding such alternative solutions, the non-optimal solutions of interest for constrained optimization models, the SoI problem. SoI problems subsume finding feasible solutions of interest (FoIs) and infeasible solutions of interest (IoIs). In all cases, the interest addressed is post-solution analysis in one form or another. Post-solution analysis of a constrained optimization model occurs after the model has been solved and a good or optimal solution for it has been found. At this point, sensitivity analysis and other questions of import for decision making come into play and for this purpose the SoIs can be very valuable. An evolutionary computation approach (in particular, a population-based metaheuristic) is proposed for solving the SoI problem and a systematic approach with a feasible-infeasible- two-population genetic algorithm is demonstrated. In this study, the effectiveness of the proposed approach on finding SoIs is demonstrated with generalized assignment problems and generalized quadratic assignment problems. Also, the applications of the proposed approach on the multi-objective optimization and robust-optimization issues are examined and illustrated with two-sided matching problems and flowshop scheduling problems respectively

Robustness maximization of parallel multichannel systems

Bit error rate (BER) minimization and SNR-gap maximization, two robustness optimization problems, are solved, under average power and bit-rate constraints, according to the waterfilling policy. Under peak-power constraint the solutions differ and this paper gives bit-loading solutions of both robustness optimization problems over independent parallel channels. The study is based on analytical approach with generalized Lagrangian relaxation tool and on greedy-type algorithm approach. Tight BER expressions are used for square and rectangular quadrature amplitude modulations. Integer bit solution of analytical continuous bit-rates is performed with a new generalized secant method. The asymptotic convergence of both robustness optimizations is proved for both analytical and algorithmic approaches. We also prove that, in conventional margin maximization problem, the equivalence between SNR-gap maximization and power minimization does not hold with peak-power limitation. Based on a defined dissimilarity measure, bit-loading solutions are compared over power line communication channel for multicarrier systems. Simulation results confirm the asymptotic convergence of both allocation policies. In non asymptotic regime the allocation policies can be interchanged depending on the robustness measure and the operating point of the communication system. The low computational effort of the suboptimal solution based on analytical approach leads to a good trade-off between performance and complexity.Comment: 27 pages, 8 figures, submitted to IEEE Trans. Inform. Theor

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