arXiv subjects

Francisco Trespalacios

Publications and source records attributed to Francisco Trespalacios.

5 recordsLinked to original sources

arXiv2025

Pseudo basic steps: Bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming

An elementary, but fundamental, operation in disjunctive programming is a basic step, which is the intersection of two disjunctions to form a new disjunction. Basic steps bring a disjunctive set in regular form closer to its disjunctive normal form and, in turn, produce relaxations that are at least as tight. An open question is: What are guaranteed bounds on the improvement from a basic step? In this paper, using properties of a convex disjunctive program's hull reformulation and multipliers from Lagrangian decomposition, we introduce an operation called a pseudo basic step and use it to provide provable bounds on this improvement along with techniques to exploit this information when solving a disjunctive program as a convex MINLP. Numerical examples illustrate the practical benefits of these bounds. In particular, on a set of K-means clustering instances, we make significant bound improvements relative to state-of-the-art commercial mixed-integer programming solvers.

math.OC↗

arXiv2023

Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

Solution methods for generalized Nash equilibrium have been dominated by variational inequalities and complementarity problems. Since these approaches fundamentally rely on the sufficiency of first-order optimality conditions for the players' decision problems, they only apply as heuristic methods when the players are modeled by nonconvex optimization problems. In contrast, this work approaches generalized Nash equilibrium using theory and methods for the global optimization of nonconvex bilevel programs. Through this perspective, we draw precise connections between generalized Nash equilibria, feasibility for bilevel programming, the Nikaido-Isoda function, and classic arguments involving Lagrangian duality and spatial price equilibrium. 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 propose a specific algorithm and show that this method can find a pure Nash equilibrium even when the players are modeled by mixed-integer programs.

math.OC↗

arXiv2022

Pooling problems under perfect and imperfect competition

We investigate pooling problems in which multiple players vie with one another to maximize individual profit in a non-cooperative competitive market. This competitive setting is interesting and worthy of study because the majority of prevailing process systems engineering models largely overlook the non-cooperative strategies that exist in real-world markets. In this work, each player controls a processing network involving intermediate tanks (or pools) where raw materials are blended together before being further combined into final products. Each player then solves a pure or mixed-integer bilinear optimization problem whose profit is influenced by other players. We present several bilevel formulations and numerical results of a novel decomposition algorithm.

math.OC↗

arXiv2020

A Note on Solving Discretely-Constrained Nash-Cournot Games via Complementarity

Discretely-constrained Nash-Cournot games have attracted attention as they arise in various competitive energy production settings in which players must make one or more discrete decisions. Gabriel et al. ["Solving discretely-constrained Nash-Cournot games with an application to power markets." Networks and Spatial Economics 13(3), 2013] claim that the set of equilibria to a discretely-constrained Nash-Cournot game coincides with the set of solutions to a corresponding discretely-constrained mixed complementarity problem. We show that this claim is false.

econ.TH↗

arXiv2019

A note on semi-infinite program bounding methods

Semi-infinite programs are a class of mathematical optimization problems with a finite number of decision variables and infinite constraints. As shown by Blankenship and Falk (Blankenship and Falk. "Infinitely constrained optimization problems." Journal of Optimization Theory and Applications 19.2 (1976): 261-281.), a sequence of lower bounds which converges to the optimal objective value may be obtained with specially constructed finite approximations of the constraint set. In (Mitsos. "Global optimization of semi-infinite programs via restriction of the right-hand side." Optimization 60.10-11 (2011): 1291-1308.), it is claimed that a modification of this lower bounding method involving approximate solution of the lower-level program yields convergent lower bounds. We show with a counterexample that this claim is false, and discuss what kind of approximate solution of the lower-level program is sufficient for correct behavior.

math.OC↗