Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

This paper introduces the concept of optimization equilibrium as an equivalently versatile definition of a generalized Nash equilibrium for multi-agent non-cooperative games. Through this modified definition of equilibrium, we draw precise connections between generalized Nash equilibria, feasibility for bilevel programming, the Nikaido-Isoda function, and classic arguments involving Lagrangian duality and social welfare maximization. Significantly, this is all in a general setting without the assumption of convexity. Along the way, we introduce the idea of minimum disequilibrium as a solution concept that reduces to traditional equilibrium when equilibrium exists. The connections with bilevel programming and related semi-infinite programming permit us to adapt global optimization methods for those classes of problems, such as constraint generation or cutting plane methods, to the problem of finding a minimum disequilibrium solution. We show that this method works, both theoretically and with a numerical example, even when the agents are modeled by mixed-integer programs

Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs---in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification.Comment: 42 pages, 10 figure

Disjunctive Aspects in Generalized Semi-infinite Programming

In this thesis the close relationship between generalized semi-infinite problems (GSIP) and disjunctive problems (DP) is considered. We start with the description of some optimization problems from timber industry and illustrate how GSIPs and DPs arise naturally in that field. Three different applications are reviewed. Next, theory and solution methods for both types of problems are examined. We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. Applying existing lower level reformulations for the obtained semi-infinite program we derive conjunctive nonlinear problems without any logical expressions, which can be locally solved by standard nonlinear solvers. In addition to this local solution procedure we propose a new branch-and-bound framework for global optimization of disjunctive programs. In contrast to the widely used reformulation as a mixed-integer program, we compute the lower bounds and evaluate the logical expression in one step. Thus, we reduce the size of the problem and work exclusively with continuous variables, which is computationally advantageous. In contrast to existing methods in disjunctive programming, none of our approaches expects any special formulation of the underlying logical expression. Where applicable, under slightly stronger assumptions, even the use of negations and implications is allowed. Our preliminary numerical results show that both procedures, the reformulation technique as well as the branch-and-bound algorithm, are reasonable methods to solve disjunctive optimization problems locally and globally, respectively. In the last part of this thesis we propose a new branch-and-bound algorithm for global minimization of box-constrained generalized semi-infinite programs. It treats the inherent disjunctive structure of these problems by tailored lower bounding procedures. Three different possibilities are examined. The first one relies on standard lower bounding procedures from conjunctive global optimization. The second and the third alternative are based on linearization techniques by which we derive linear disjunctive relaxations of the considered sub-problems. Solving these by either mixed-integer linear reformulations or, alternatively, by disjunctive linear programming techniques yields two additional possibilities. Our numerical results on standard test problems with these three lower bounding procedures show the merits of our approach

Particle swarm optimization for bi-level pricing problems in supply chains

With rapid technological innovation and strong competition in hi-tech industries such as computer and communication organizations, the upstream component price and the downstream product cost usually decline significantly with time. As a result, an effective pricing supply chain model is very important. This paper first establishes two bi-level pricing models for pricing problems with the buyer and the vendor in a supply chain designated as the leader and the follower, respectively. A particle swarm optimization (PSO) based algorithm is developed to solve problems defined by these bi-level pricing models. Experiments illustrate that this PSO based algorithm can achieve a profit increase for buyers or vendors if they are treated as the leaders under some situations, compared with the existing methods. © 2010 Springer Science+Business Media, LLC

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

Theoretical and numerical analysis of optimization problems with applications to continuum mechanics

Tese de doutoramento, Matemática (Análise Numérica e Matemática Computacional), Universidade de Lisboa, Faculdade de Ciências, 2013Fundação para a Ciência e a Tecnologia (FCT, Financiamento Base 2010 - ISFL/1/209, SFRH/BD/44343/2008); Mathematical and Numerical Methods in Mechanics research group do Centro de Matemática e Aplicações Fundamental da U

On Control and Estimation of Large and Uncertain Systems

This thesis contains an introduction and six papers about the control and estimation of large and uncertain systems. The first paper poses and solves a deterministic version of the multiple-model estimation problem for finite sets of linear systems. The estimate is an interpolation of Kalman filter estimates. It achieves a provided energy gain bound from disturbances to the point-wise estimation error, given that the gain bound is feasible. The second paper shows how to compute upper and lower bounds for the smallest feasible gain bound. The bounds are computed via Riccati recursions. The third paper proves that it is sufficient to consider observer-based feedback in output-feedback control of linear systems with uncertain parameters, where the uncertain parameters belong to a finite set. The paper also contains an example of a discrete-time integrator with unknown gain. The fourth paper argues that the current methods for analyzing the robustness of large systems with structured uncertainty do not distinguish between sparse and dense perturbations and proposes a new robustness measure that captures sparsity. The paper also thoroughly analyzes this new measure. In particular, it proposes an upper bound that is amenable to distributed computation and valuable for control design. The fifth paper solves the problem of localized state-feedback L2 control with communication delay for large discrete-time systems. The synthesis procedure can be performed for each node in parallel. The paper combines the localized state-feedback controller with a localized Kalman filter to synthesize a localized output feedback controller that stabilizes the closed-loop subject to communication constraints. The sixth paper concerns optimal linear-quadratic team-decision problems where the team does not have access to the model. Instead, the players must learn optimal policies by interacting with the environment. The paper contains algorithms and regret bounds for the first- and zeroth-order information feedback