A hybrid heuristic for solving mixed integer nonlinear programming problems

Orientadores: Márcia Aparecida Gomes Ruggiero, Antonio Carlos MorettiTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo neste trabalho é abordar problemas formulados como MINLP (Mixed Integer Nonlinear Programming). Propomos um método de resolução heurístico baseado em ideias de métodos do tipo Restauração Inexata combinado com a heurística denominada Feasibility Pump. Os métodos de Restauração Inexata foram propostos para resolução de problemas não lineares com variáveis contínuas. As iterações envolvem duas fases, Restauração (fase da viabilidade) e Otimalidade. A heurística Feasibility Pump foi proposta para obter soluções factíveis para problemas de otimização com variáveis inteiras, MILPs (Mixed Integer Linear Programming) e MINLPs. Neste trabalho adaptamos as duas fases dos métodos de Restauração Inexata ao contexto de problemas com variáveis inteiras, MINLP, buscando avanços na viabilidade (fase da Restauração) através da heurística Feasibility Pump. Na fase de otimalidade resolvemos dois subproblemas, no primeiro a condição de integralidade sobre as variáveis é relaxada e construímos um PNL (Problema de Programação Não Linear), no segundo as restrições não lineares são relaxadas e construímos um MILP. Um processo mestre coordena os subproblemas que são resolvidos em cada fase. O desempenho do algoritmo foi analisado e validado através da resolução de um conjunto clássico de problemasAbstract: The aim of this work is to address problems formulated as MINLP (Mixed Integer Nonlinear Programming). We propose a heuristic resolution method based on Inexact Restoration methods combined with the Feasibility Pump heuristic. The Inexact Restoration methods were proposed for solving nonlinear problems with continuous variables. These methods involve two phases, Restoration (viability phase) and Optimality. The Feasibility Pump heuristic was proposed to obtain feasible solutions for optimization problems with integer variables, MILPs (Mixed Integer Linear Programming) and MINLPs. In this work we adapt the two phases of the Inexact Restoration method in the context of problems with integer variables, MINLP, seeking advances in feasibility (Restoration phase) through the Feasibility Pump heuristic. In the optimality phase, two subproblems are solved, in the first the integrality constraints are relaxed and we construct a NLP (Nonlinear Programming), in the second the nonlinear constraints are relaxed and we construct a MILP. A master process coordinates the subproblems to be solved at each stage. The performance of the final algorithm was analised in a set of classical problemsDoutoradoMatematica AplicadaDoutora em Matemática Aplicada2013/21515-9FAPESPCAPE

Ten years of feasibility pump, and counting

The Feasibility Pump (fp) is probably the best-known primal heuristic for mixed-integer programming. The original work by Fischetti et al. (Math Program 104(1):91\u2013104, 2005), which introduced the heuristic for 0\u20131 mixed-integer linear programs, has been succeeded by more than twenty follow-up publications which improve the performance of the fp and extend it to other problem classes. Year 2015 was the tenth anniversary of the first fp publication. The present paper provides an overview of the diverse Feasibility Pump literature that has been presented over the last decade

Proximal-like algorithms for equilibrium seeking in mixed-integer Nash equilibrium problems

We consider potential games with mixed-integer variables, for which we propose two distributed, proximal-like equilibrium seeking algorithms. Specifically, we focus on two scenarios: i) the underlying game is generalized ordinal and the agents update through iterations by choosing an exact optimal strategy; ii) the game admits an exact potential and the agents adopt approximated optimal responses. By exploiting the properties of integer-compatible regularization functions used as penalty terms, we show that both algorithms converge to either an exact or an $\epsilon$-approximate equilibrium. We corroborate our findings on a numerical instance of a Cournot oligopoly model

Tailored Presolve Techniques in Branch-and-Bound Method for Fast Mixed-Integer Optimal Control Applications

Mixed-integer model predictive control (MI-MPC) can be a powerful tool for modeling hybrid control systems. In case of a linear-quadratic objective in combination with linear or piecewise-linear system dynamics and inequality constraints, MI-MPC needs to solve a mixed-integer quadratic program (MIQP) at each sampling time step. This paper presents a collection of block-sparse presolve techniques to efficiently remove decision variables, and to remove or tighten inequality constraints, tailored to mixed-integer optimal control problems (MIOCP). In addition, we describe a novel heuristic approach based on an iterative presolve algorithm to compute a feasible but possibly suboptimal MIQP solution. We present benchmarking results for a C code implementation of the proposed BB-ASIPM solver, including a branch-and-bound (B&B) method with the proposed tailored presolve techniques and an active-set based interior point method (ASIPM), compared against multiple state-of-the-art MIQP solvers on a case study of motion planning with obstacle avoidance constraints. Finally, we demonstrate the computational performance of the BB-ASIPM solver on the dSPACE Scalexio real-time embedded hardware using a second case study of stabilization for an underactuated cart-pole with soft contacts.Comment: 27 pages, 7 figures, 2 tables, submitted to journal of Optimal Control Applications and Method

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Advances in Energy System Optimization

The papers presented in this open access book address diverse challenges in decarbonizing energy systems, ranging from operational to investment planning problems, from market economics to technical and environmental considerations, from distribution grids to transmission grids, and from theoretical considerations to data provision concerns and applied case studies. While most papers have a clear methodological focus, they address policy-relevant questions at the same time. The target audience therefore includes academics and experts in industry as well as policy makers, who are interested in state-of-the-art quantitative modelling of policy relevant problems in energy systems. The 2nd International Symposium on Energy System Optimization (ISESO 2018) was held at the Karlsruhe Institute of Technology (KIT) under the symposium theme “Bridging the Gap Between Mathematical Modelling and Policy Support” on October 10th and 11th 2018. ISESO 2018 was organized by the KIT, the Heidelberg Institute for Theoretical Studies (HITS), the Heidelberg University, the German Aerospace Center and the University of Stuttgart

Outer-approximation algorithms for nonsmooth convex MINLP problems with chance constraints

Orientador: Prof. Dr. Yuan Jin YunCoorientador: Prof. Dr. Welington Luis de OliveiraTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Matemática. Defesa : Curitiba, 13/04/2018Inclui referências: p.82-88Resumo: As restri?c˜oes de probabilidade desempenham um papel fundamental nos problemas de otimiza?c˜ao envolvendo incertezas. Essas restri?c˜oes exigem que um sistema de desigualdade dependendo de um vetor aleat'orio tenha que ser satisfeito com uma probabilidade suficientemente alta. Neste trabalho, lidamos com problemas de otimiza?c˜ao com restri?c˜oes de probabilidades envolvendo vari'aveis inteiras. Assumimos que as fun?c˜oes envolvidas s˜ao convexas e a restri?c˜ao de probabilidade tenha propriedade generalizada de convexidade. Para lidar com problemas de otimiza?c˜ao desse tipo, combinamos o algoritmo de aproxima ?c˜ao externa (OA) e o algoritmo de feixes. Os algoritmos OA tem sido aplicado para problemas su'aveis e para uma pequena classe limitada de problemas n˜ao-su'aveis. Neste trabalho, estendemos o algoritmo OA para lidar com problemas mais gerais n˜ao-su'aveis. Al'em disso, mostramos que quando os subproblemas n˜ao-lineares resultantes do algoritmo OA s˜ao resolvidos por um m'etodo de feixes, ent˜ao os subgradientes que satisfazem as condi?c˜oes de Karush Kuhn Tucker (KKT) est˜ao prontamente dispon'?veis independentemente da estrutura das fun?c˜oes convexas n˜ao-su'aveis. Esta propriedade 'e crucial para provar a convergˆencia (finita) do algoritmo OA. Problemas com restri?c˜oes probabil'?sticas aparecem, por exemplo, em modelos de energia (estoc'asticos). No contexto de interesse, pelo menos uma das restri?c˜oes n˜ao lineares envolve uma fun?c˜ao de probabilidade P[h(x, y) ? ?], onde h 'e uma fun?c˜ao cˆoncava e ? ? Rm 'e um vetor aleat'orio. Em geral, uma integra?c˜ao num'erica multidimensional 'e empregada para avaliar essa fun?c˜ao de probabilidade. Como uma alternativa para lidar com restri?c˜oes de probabilidades (que 'e muito cara computacionalmente), propomos a aproxima?c˜ao da medida de probabilidade P por uma c'opula apropriada. N'os investigamos uma fam'?lia de c'opulas n˜ao-su'aveis e fornecemos algumas propriedades generalizadas de convexidade novas e 'uteis. Em particular, provamos que a fam'?lia de c'opulas de Zhang 'e ??cˆoncava para todo ? ? 0. Esse resultado nos permite aproximar as restri?c˜oes probabil'?sticas por restri?c˜oes muito mais simples envolvendo c'opulas. Avaliamos numericamente as abordagens dadas em duas classe de problemas provenientes do gerenciamento do sistema de energia el'etrica. Palavras-chave: Otimiza¸c˜ao n˜ao-linear inteira, Otimiza¸c˜ao Estoc'astica, Restri¸c˜oes Probabil '?sticas.Abstract: Probability constraints play a key role in optimization problems involving uncertainties. These constraints (also known as chance constraints) require that an inequality system depending on a random vector has to be satisfied with high enough probability. In this work we deal with chance-constrained optimization problems having mixed-integer variables. We assume that the involved functions are convex and the probability constraint has generalized convexity properties. In order to deal with optimization problems of this type, we combine outer-approximation (OA) and bundle method algorithms. OA algorithms have been applied to smooth problems and to a small class of nonsmooth problems. In this work we extend the OA to handle more general nonsmooth problems. Moreover, we show that when the resulting OA's nonlinear subproblems are solved by a bundle method, then subgradients satisfying the Karush-Kuhn-Tucker (KKT) conditions are readily available regardless the structure of the nonsmooth convex functions. This property is crucial for proving (finite) convergence of the OA algorithm. Chance-constrained problems appear, for instance, in (stochastic) energy models. In the context of interest, at least one nonlinear constraint models the probability function P[h(x, y) ? ?], where h is a concave map and ? ? Rm is a random vector. In general, multidimensional numerical integration is employed to evaluate this probability function. As an alternative to deal with probability constraints (which is very expensive computationally), we propose approximating the probability measure P with a suitable copula. We investigate a family of nonsmooth copulae and provide some new and useful generalized convexity properties. In particular, we prove that Zhang's copulae are ?-concave for all ? ? 0. This result allows us to approximate chance-constrained programs by much simpler copula-constrained ones. We assess numerically the given approaches on two classes of problems coming from power system management. Keywords: Mixed-Integer Nonlinear Optimization, Stochastic Optimization, Chance constraints