Optimization and Equilibrium Problems with Equilibrium Constraints in Infinite-Dimensional Spaces

The paper is devoted to applications of modern variational f).nalysis to the study of constrained optimization and equilibrium problems in infinite-dimensional spaces. We pay a particular attention to the remarkable classes of optimization and equilibrium problems identified as MPECs (mathematical programs with equilibrium constraints) and EPECs (equilibrium problems with equilibrium constraints) treated from the viewpoint of multiobjective optimization. Their underlying feature is that the major constraints are governed by parametric generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and can be handled by using an appropriate machinery of generalized differentiation exhibiting a rich/full calculus. The case of infinite-dimensional spaces is significantly more involved in comparison with finite dimensions, requiring in addition a certain sufficient amount of compactness and an efficient calculus of the corresponding sequential normal compactness (SNC) properties

Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics

Gridless Evolutionary Approach for Line Spectral Estimation with Unknown Model Order

Gridless methods show great superiority in line spectral estimation. These methods need to solve an atomic $l_0$ norm (i.e., the continuous analog of $l_0$ norm) minimization problem to estimate frequencies and model order. Since this problem is NP-hard to compute, relaxations of atomic $l_0$ norm, such as nuclear norm and reweighted atomic norm, have been employed for promoting sparsity. However, the relaxations give rise to a resolution limit, subsequently leading to biased model order and convergence error. To overcome the above shortcomings of relaxation, we propose a novel idea of simultaneously estimating the frequencies and model order by means of the atomic $l_0$ norm. To accomplish this idea, we build a multiobjective optimization model. The measurment error and the atomic $l_0$ norm are taken as the two optimization objectives. The proposed model directly exploits the model order via the atomic $l_0$ norm, thus breaking the resolution limit. We further design a variable-length evolutionary algorithm to solve the proposed model, which includes two innovations. One is a variable-length coding and search strategy. It flexibly codes and interactively searches diverse solutions with different model orders. These solutions act as steppingstones that help fully exploring the variable and open-ended frequency search space and provide extensive potentials towards the optima. Another innovation is a model order pruning mechanism, which heuristically prunes less contributive frequencies within the solutions, thus significantly enhancing convergence and diversity. Simulation results confirm the superiority of our approach in both frequency estimation and model order selection.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

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

Design of of model-based controllers via parametric programming

Imperial Users onl

Data-driven Inverse Optimization with Imperfect Information

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision implied by a particular candidate objective) differs from the agent's {\em actual} response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches

Multiobjective optimization for interwoven systems

In practical situations, complex systems are often composed of subsystems or subproblems with single or multiple objectives. These subsystems focus on different aspects of the overall system, but they often have strong interactions with each other and they are usually not sequentially ordered or obviously decomposable. Thus, the individual solutions of subproblems do not generally induce a solution for the overall system. Here, we strive to identify "re-composition architectures" of such "interwoven" systems. Our intention is to connect the subsystems adequately, analyze the resulting performance, model/solve the overall system, and improve the overall solution instead of just solving each subsystem separately. We review recent developments in this field and discuss modeling and solution paradigms in a general and unified framework using the example of an interwoven system consisting of two interacting subsystems