Farkas-Type Results for Vector-Valued Functions with Applications

The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.This research was partially supported by MINECO of Spain and FEDER of EU, Grant MTM2014-59179-C2-1-P, by the project DP160100854 from the Australian Research Council, and by the project B2015-28-04: “A new approach to some classes of optimization problems” from the Vietnam National University - HCM city, Vietnam

Applying multiobjective evolutionary algorithms in industrial projects

During the recent years, multiobjective evolutionary algorithms have matured as a flexible optimization tool which can be used in various areas of reallife applications. Practical experiences showed that typically the algorithms need an essential adaptation to the specific problem for a successful application. Considering these requirements, we discuss various issues of the design and application of multiobjective evolutionary algorithms to real-life optimization problems. In particular, questions on problem-specific data structures and evolutionary operators and the determination of method parameters are treated. As a major issue, the handling of infeasible intermediate solutions is pointed out. Three application examples in the areas of constrained global optimization (electronic circuit design), semi-infinite programming (design centering problems), and discrete optimization (project scheduling) are discussed

Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators $A=\partial\Phi+B$, where $\partial\Phi$ is the subdifferential of a convex lower semicontinuous function $\Phi$, and $B$ is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by $A$, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multi-criteria decision processes.Comment: 25 page

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Barzilai-Borwein Descent Methods for Multiobjective Optimization Problems with Variable Trade-off Metrics

The imbalances and conditioning of the objective functions influence the performance of first-order methods for multiobjective optimization problems (MOPs). The latter is related to the metric selected in the direction-finding subproblems. Unlike single-objective optimization problems, capturing the curvature of all objective functions with a single Hessian matrix is impossible. On the other hand, second-order methods for MOPs use different metrics for objectives in direction-finding subproblems, leading to a high per-iteration cost. To balance per-iteration cost and better curvature exploration, we propose a Barzilai-Borwein descent method with variable metrics (BBDMO\_VM). In the direction-finding subproblems, we employ a variable metric to explore the curvature of all objectives. Subsequently, Barzilai-Borwein's method relative to the variable metric is applied to tune objectives, which mitigates the effect of imbalances. We investigate the convergence behaviour of the BBDMO\_VM, confirming fast linear convergence for well-conditioned problems relative to the variable metric. In particular, we establish linear convergence for problems that involve some linear objectives. These convergence results emphasize the importance of metric selection, motivating us to approximate the trade-off of Hessian matrices to better capture the geometry of the problem. Comparative numerical results confirm the efficiency of the proposed method, even when applied to large-scale and ill-conditioned problems

MULTIOBJECTIVE AND MULTISCALE OPTIMIZATION OF COMPOSITE MATERIALS BY MEANS OF EVOLUTIONARY COMPUTATIONS

The paper deals with the multiobjective and multiscale optimization of heterogeneous structures by means of computational intelligence methods. The aim of the paper is to find optimal properties of composite structures in a macro scale modifying their microstructure. At least two contradictory optimization criteria are considered simultaneously. A numerical homogenization concept with a representative volume element is applied to obtain equivalent macro-scale elastic constants. An in-house multiobjective evolutionary algorithm MOOPTIM is applied to solve the considered optimization tasks. The finite element method is used to solve the boundary-value problem in both scales. A numerical example is attached

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented