Abstract: We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages. We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems. Our constructs enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of sophisticated structures such as shared constraints and additional constraints on their solutions. We define shared variables and demonstrate their uses for sparse reformulation, economic equilibrium problems sharing economic states, mixed pricing behavior of agents, and so on. We give some equilibrium and variational examples from the literature and describe how to formulate them using our framework. Experimental results comparing performance of various complementarity formulations for shared variables are provided. Our framework has been implemented and is available within GAMS/EMP. PubDate: 2019-03-02
Abstract: We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by differential equations, i.e. optimal control problems. CasADi is written in self-contained C++, but is most conveniently used via full-featured interfaces to Python, MATLAB or Octave. Since its inception in late 2009, it has been used successfully for academic teaching as well as in applications from multiple fields, including process control, robotics and aerospace. This article gives an up-to-date and accessible introduction to the CasADi framework, which has undergone numerous design improvements over the last 7 years. PubDate: 2019-03-01
Abstract: The handling of symmetries in mixed integer programs in order to speed up the solution process of branch-and-cut solvers has recently received significant attention, both in theory and practice. This paper compares different methods for handling symmetries using a common implementation framework. We start by investigating the computation of symmetries and analyze the symmetries present in the MIPLIB 2010 instances. It turns out that many instances are affected by symmetry and most symmetry groups contain full symmetric groups as factors. We then present (variants of) six symmetry handling methods from the literature. Their implementation is tested on several testsets. On very symmetric instances used previously in the literature, it is essential to use methods like isomorphism pruning, orbital fixing, or orbital branching. Moreover, tests on the MIPLIB instances show that isomorphism pruning, orbital fixing, or adding symmetry breaking inequalities allow to speed-up the solution process by about 15% and more instances can be solved within the time limit. PubDate: 2019-03-01
Abstract: We consider a quadratic program with a few negative eigenvalues (QP-r-NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified \(\epsilon \) -tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within \(\epsilon \) -tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP-r-NE instances when \(r\le 10\) , and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739. PubDate: 2019-03-01
Abstract: Nonconvex mixed-binary nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use an iterative solution procedure for solving series of regularized problems. In the case of success, these procedures result in a feasible solution of the original mixed-binary nonlinear problem. Since we rely on local nonlinear programming solvers the resulting method is fast and we further improve its reliability by additional algorithmic techniques. We show the strength of our method by an extensive computational study on 662 MINLPLib2instances, where our methods are able to produce feasible solutions for \({60}{\%}\) of all instances in at most \({10}\,{\hbox {s}}\) . PubDate: 2019-03-01
Abstract: A (convex) polytope P is said to be 2-level if for each hyperplane H that supports a facet of P, the vertices of P can be covered with H and exactly one other translate of H. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for \(d \leqslant 7\) . Our approach is inductive: for each fixed \((d-1)\) -dimensional 2-level polytope \(P_0\) , we enumerate all d-dimensional 2-level polytopes P that have \(P_0\) as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet \(P_0\) , we obtain all 2-level polytopes in dimension d. PubDate: 2019-03-01
Abstract: Quadratic optimization problems (QPs) are ubiquitous, and solution algorithms have matured to a reliable technology. However, the precision of solutions is usually limited due to the underlying floating-point operations. This may cause inconveniences when solutions are used for rigorous reasoning. We contribute on three levels to overcome this issue. First, we present a novel refinement algorithm to solve QPs to arbitrary precision. It iteratively solves refined QPs, assuming a floating-point QP solver oracle. We prove linear convergence of residuals and primal errors. Second, we provide an efficient implementation, based on SoPlex and qpOASES that is publicly available in source code. Third, we give precise reference solutions for the Maros and Mészáros benchmark library. PubDate: 2019-02-06
Abstract: This paper addresses the sparse principal component analysis (SPCA) problem for covariance matrices in dimension n aiming to find solutions with sparsity k using mixed integer optimization. We propose a tailored branch-and-bound algorithm, Optimal-SPCA, that enables us to solve SPCA to certifiable optimality in seconds for \(n = 100\) s, \(k=10\) s. This same algorithm can be applied to problems with \(n=10{,}000\,\mathrm{s}\) or higher to find high-quality feasible solutions in seconds while taking several hours to prove optimality. We apply our methods to a number of real data sets to demonstrate that our approach scales to the same problem sizes attempted by other methods, while providing superior solutions compared to those methods, explaining a higher portion of variance and permitting complete control over the desired sparsity. The software that was reviewed as part of this submission has been given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.2027898. PubDate: 2019-01-01
Abstract: We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In order to overcome this shortcoming, we reformulate the problem using the perspective of the quadratic function. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve convex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed algorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they provide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LP-based branch-and-bound algorithm with extended formulations. The software that was reviewed as part of this submission was given the Digital Object identifier https://doi.org/10.5281/zenodo.1489153. PubDate: 2018-12-03
Abstract: Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today’s power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of the involved Hermitian matrix variable and characterize all such minors into three types. We show the equivalence of these minor constraints to the physical constraints of voltage angle differences summing to zero over three- and four-cycles in the power network. We study second-order conic programming (SOCP) relaxations of this minor reformulation and propose strong cutting planes, convex envelopes, and bound tightening techniques to strengthen the resulting SOCP relaxations. We then propose an SOCP-based spatial branch-and-cut method to obtain the global optimum of AC OPF. Extensive computational experiments show that the proposed algorithm significantly outperforms the state-of-the-art SDP-based OPF solver and on a simple personal computer is able to obtain on average a \(0.71\%\) optimality gap in no more than 720 s for the most challenging power system instances in the literature. PubDate: 2018-12-01
Abstract: We consider the problem of optimizing an unknown function given as an oracle over a mixed-integer box-constrained set. We assume that the oracle is expensive to evaluate, so that estimating partial derivatives by finite differences is impractical. In the literature, this is typically called a black-box optimization problem with costly evaluation. This paper describes the solution methodology implemented in the open-source library RBFOpt, available on COIN-OR. The algorithm is based on the Radial Basis Function method originally proposed by Gutmann (J Glob Optim 19:201–227, 2001. https://doi.org/10.1023/A:1011255519438), which builds and iteratively refines a surrogate model of the unknown objective function. The two main methodological contributions of this paper are an approach to exploit a noisy but less expensive oracle to accelerate convergence to the optimum of the exact oracle, and the introduction of an automatic model selection phase during the optimization process. Numerical experiments show that RBFOpt is highly competitive on a test set of continuous and mixed-integer nonlinear unconstrained problems taken from the literature: it outperforms the open-source solvers included in our comparison by a large amount, and performs slightly better than a commercial solver. Our empirical evaluation provides insight on which parameterizations of the algorithm are the most effective in practice. The software reviewed as part of this submission was given the Digital Object Identifier ( DOI ) https://doi.org/10.5281/zenodo.597767. PubDate: 2018-12-01
Abstract: Semidefinite programming, SDP, relaxations have proven to be extremely strong for many hard discrete optimization problems. This is in particular true for the quadratic assignment problem, QAP, arguably one of the hardest NP-hard discrete optimization problems. There are several difficulties that arise in efficiently solving the SDP relaxation, e.g., increased dimension; inefficiency of the current primal–dual interior point solvers in terms of both time and accuracy; and difficulty and high expense in adding cutting plane constraints. We propose using the alternating direction method of multipliers ADMM in combination with facial reduction, FR, to solve the SDP relaxation. This first order approach allows for: inexpensive iterations, a method of cheaply obtaining low rank solutions; and a trivial way of exploiting the FR for adding cutting plane inequalities. In fact, we solve the doubly nonnegative, DNN, relaxation that includes both the SDP and all the nonnegativity constraints. When compared to current approaches and current best available bounds we obtain robustness, efficiency and improved bounds. PubDate: 2018-12-01
Abstract: We introduce a mathematical programming approach to building rule lists, which are a type of interpretable, nonlinear, and logical machine learning classifier involving IF-THEN rules. Unlike traditional decision tree algorithms like CART and C5.0, this method does not use greedy splitting and pruning. Instead, it aims to fully optimize a combination of accuracy and sparsity, obeying user-defined constraints. This method is useful for producing non-black-box predictive models, and has the benefit of a clear user-defined tradeoff between training accuracy and sparsity. The flexible framework of mathematical programming allows users to create customized models with a provable guarantee of optimality. The software reviewed as part of this submission was given the DOI (Digital Object Identifier) https://doi.org/10.5281/zenodo.1344142. PubDate: 2018-12-01
Abstract: Consensus decision-making, a widely utilized group decision-making process, requires the consent of all participants. We consider consensus stopping games, a class of stochastic games arising in the context of consensus decision-making that require the consent of all players to terminate the game. We show that a consensus stopping game may have many pure stationary equilibria, which in turn raises the question of equilibrium selection. Given an objective criterion, we study the NP-hard problem of finding a best pure stationary equilibrium. We characterize the pure stationary equilibria, show that they form an independence system, and develop several families of valid inequalities. We then solve the equilibrium selection problem as a mixed-integer linear program by a branch-and-cut approach. Our computational results demonstrate the effectiveness of our approach over a commercial solver. PubDate: 2018-11-02
Abstract: A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a modeling paradigm for a broad collection of problems, including bilevel programs, Stackelberg games, inverse quadratic programs, and problems involving equilibrium constraints. The presence of the complementarity constraints results in a nonconvex optimization problem. We develop a branch-and-cut algorithm to find a global optimum for this class of optimization problems, where we branch directly on complementarities. We develop branching rules and feasibility recovery procedures and demonstrate their computational effectiveness in a comparison with CPLEX. The implementation builds on CPLEX through the use of callback routines. The computational results show that our approach is a strong alternative to constructing an integer programming formulation using big-M terms to represent bounds for variables, with testing conducted on general LPCCs as well as on instances generated from bilevel programs with convex quadratic lower level problems. PubDate: 2018-09-27
Abstract: This paper describes a new instance library for quadratic programming (QP), i.e., the family of continuous and (mixed)-integer optimization problems where the objective function and/or the constraints are quadratic. QP is a very diverse class of problems, comprising sub-classes ranging from trivial to undecidable. This diversity is reflected in the variety of QP solution methods, ranging from entirely combinatorial approaches to completely continuous algorithms, including many methods for which both aspects are fundamental. Selecting a set of instances of QP that is at the same time not overwhelmingly onerous but sufficiently challenging for the different, interested communities is therefore important. We propose a simple taxonomy for QP instances leading to a systematic problem selection mechanism. We then briefly survey the field of QP, giving an overview of theory, methods and solvers. Finally, we describe how the library was put together, and detail its final contents. PubDate: 2018-09-22
Authors:Xudong Li; Defeng Sun; Kim-Chuan Toh Abstract: In this paper, we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality and inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite cone constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point for the second phase. In QSDPNAL-Phase II, we design an augmented Lagrangian method (ALM) wherein the inner subproblem in each iteration is solved via inexact semismooth Newton based algorithms. Simple and implementable stopping criteria are designed for the ALM. Moreover, under mild conditions, we are able to establish the rate of convergence of the proposed algorithm and prove the R-(super)linear convergence of the KKT residual. In the implementation of QSDPNAL, we also develop efficient techniques for solving large scale linear systems of equations under certain subspace constraints. More specifically, simpler and yet better conditioned linear systems are carefully designed to replace the original linear systems and novel shadow sequences are constructed to alleviate the numerical difficulties brought about by the crucial subspace constraints. Extensive numerical results for various large scale QSDPs show that our two-phase algorithm is highly efficient and robust in obtaining accurate solutions. The software reviewed as part of this submission was given the DOI (Digital Object Identifier) https://doi.org/10.5281/zenodo.1206980. PubDate: 2018-04-13
Authors:Daniel Juhl; David M. Warme; Pawel Winter; Martin Zachariasen Abstract: The GeoSteiner software package has for about 20 years been the fastest (publicly available) program for computing exact solutions to Steiner tree problems in the plane. The computational study by Warme, Winter and Zachariasen, published in 2000, documented the performance of the GeoSteiner approach—allowing the exact solution of Steiner tree problems with more than a thousand terminals. Since then, a number of algorithmic enhancements have improved the performance of the software package significantly. We describe these (previously unpublished) enhancements, and present a new computational study wherein we run the current code on the largest problem instances from the 2000-study, and on a number of larger problem instances. The computational study is performed using the commercial GeoSteiner 4.0 code base, and the performance is compared to the publicly available GeoSteiner 3.1 code base as well as the code base from the 2000-study. The software studied in the paper is being released as GeoSteiner 5.0 under an open source license. PubDate: 2018-02-20 DOI: 10.1007/s12532-018-0135-8
Authors:Hande Y. Benson; David F. Shanno Abstract: Quasi-Newton methods based on the symmetric rank-one (SR1) update have been known to be fast and provide better approximations of the true Hessian than popular rank-two approaches, but these properties are guaranteed under certain conditions which frequently do not hold. Additionally, SR1 is plagued by the lack of guarantee of positive definiteness for the Hessian estimate. In this paper, we propose cubic regularization as a remedy to relax the conditions on the proofs of convergence for both speed and accuracy and to provide a positive definite approximation at each step. We show that the n-step convergence property for strictly convex quadratic programs is retained by the proposed approach. Extensive numerical results on unconstrained problems from the CUTEr test set are provided to demonstrate the computational efficiency and robustness of the approach. PubDate: 2018-02-12 DOI: 10.1007/s12532-018-0136-7
Authors:Yunhai Xiao; Liang Chen; Donghui Li Abstract: In this paper, we propose a generalized alternating direction method of multipliers (ADMM) with semi-proximal terms for solving a class of convex composite conic optimization problems, of which some are high-dimensional, to moderate accuracy. Our primary motivation is that this method, together with properly chosen semi-proximal terms, such as those generated by the recent advance of block symmetric Gauss–Seidel technique, is capable of tackling these problems. Moreover, the proposed method, which relaxes both the primal and the dual variables in a natural way with a common relaxation factor in the interval of (0, 2), has the potential of enhancing the performance of the classic ADMM. Extensive numerical experiments on various doubly non-negative semidefinite programming problems, with or without inequality constraints, are conducted. The corresponding results showed that all these multi-block problems can be successively solved, and the advantage of using the relaxation step is apparent. PubDate: 2018-01-23 DOI: 10.1007/s12532-018-0134-9
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