Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with Equilibrium Constraints (MPECs). We first reformulate the binary program as an equivalent augmented biconvex optimization problem with a bilinear equality constraint, then we propose two penalization/regularization methods (exact penalty and alternating direction) to solve it. The resulting algorithms seek desirable solutions to the original problem via solving a sequence of linear programming convex relaxation subproblems. In addition, we prove that both the penalty function and augmented Lagrangian function, induced by adding the complementarity constraint to the objectives, are exact, i.e., they have the same local and global minima with those of the original binary program when the penalty parameter is over some threshold. The convergence of both algorithms can be guaranteed, since they essentially reduce to block coordinate descent in the literature. Finally, we demonstrate the effectiveness and versatility of our methods on several important problems, including graph bisection, constrained image segmentation, dense subgraph discovery, modularity clustering and Markov random fields. Extensive experiments show that our methods outperform existing popular techniques, such as iterative hard thresholding, linear programming relaxation and semidefinite programming relaxation.
This paper provides necessary and sufficient optimality conditions for abstract constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections.
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a hierarchy of semidefinite programming (SDP) relaxations and prove the convergence result for the method. Numerical experiments are presented to show the efficiency of the proposed algorithm.
We explore how to build a vector field from the various functions involved in a given mathematical program, and show that locally-stable equilibria of the underlying dynamical system are precisely the local solutions of the optimization problem. The general situation in which explicit inequality constraints are present is especially interesting as the vector field has to be discontinuous, and so one is led to consider discontinuous dynamical systems and their equilibria.
The concepts of infinity and infinitesimal in mathematics date back to anciens Greek and have always attracted great attention. Very recently, a new methodology has been proposed by Sergeyev for performing calculations with infinite and infinitesimal quantities, by introducing an infinite unit of measure expressed by the numeral grossone. An important characteristic of this novel approach is its attention to numerical aspects. In this paper we will present some possible applications and use of grossone in Operations Research and Mathematical Programming. In particular, we will show how the use…
We propose a new model to detect the overlapping communities of a network that is based on cooperative games and mathematical programming. More specifically, communities are defined as stable coalitions of a weighted graph community game and they are revealed as the optimal solution of a mixed-integer linear programming problem. Exact optimal solutions are obtained for small and medium sized instances and and it is shown that they provide useful information about the network structure, improving on previous contributions. Next, a heuristic algorithm is developed to solve the largest instances …
Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current …
The Piecewise Polynomial Interpolation (PPI) function approach is aimed at solving nonlinear programming problems with disjoint feasible regions. In such problems, disjointedness is generally associated with prohibited operating zones, which correspond to bands of values that a variable is not allowed to assume. An analytical implication of such prohibited operating zones is to make the objective function, as well as its domain, discontinuous. The PPI function approach consists in replacing the constraints associated with prohibited operating zones by an equivalent set of equality and inequali…
This paper considers mathematical programs, whose constraints are expressed by a parameterized vector equilibrium problem. The latter is a well recognized framework, which is able to cover multicriteria optimization, vector variational inequalities and complementarity problems. As the solutions to vector equilibrium problems are here intended in a strong sense, the consequent MPEC problems result in a class still little explored by the existing literature. Some necessary optimality conditions for such programs are established following a penalization approach. To derive and express these condi…
We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the complementarity measure and a nonlinear programming with general constraints. A strategy generalizing ideas of Byrd-Omojokun's trust region method is used to compute steps. By penalizing the tangential constraints into the objective function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like strategy is used to balance the improvements on feasibilit…
In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. We derive a new necessary optimality condition which is sharper than the usual M-stationary condition and is applicable even when no constraint qualifications hold for the corresponding mathematical program with complementarity constraints (MPCC) reformulation.
This study investigates the progress made in LP and MILP solver performance during the last two decades by comparing the solver software from the beginning of the millennium with the codes available today. On average, we found out that for solving LP/MILP, computer hardware got about 20 times faster, and the algorithms improved by a factor of about nine for LP and around 50 for MILP, which gives a total speed-up of about 180 and 1,000 times, respectively. However, these numbers have a very high variance and they considerably underestimate the progress made on the algorithmic side: many problem…
The optimal binning is the optimal discretization of a variable into bins given a discrete or continuous numeric target. We present a rigorous and extensible mathematical programming formulation for solving the optimal binning problem for a binary, continuous and multi-class target type, incorporating constraints not previously addressed. For all three target types, we introduce a convex mixed-integer programming formulation. Several algorithmic enhancements, such as automatic determination of the most suitable monotonic trend via a Machine-Learning-based classifier and implementation aspects …
Sparsity constrained minimization captures a wide spectrum of applications in both machine learning and signal processing. This class of problems is difficult to solve since it is NP-hard and existing solutions are primarily based on Iterative Hard Thresholding (IHT). In this paper, we consider a class of continuous optimization techniques based on Mathematical Programs with Equilibrium Constraints (MPECs) to solve general sparsity constrained problems. Specifically, we reformulate the problem as an equivalent biconvex MPEC, which we can solve using an exact penalty method or an alternating di…
This work is a geometrical approach to the optimization problem motivated by transportation system management. First, an attempt has been made to furnish a comprehensive account of geometric programming based on the elementary Finsler geometry in $I\!\!R^n$. Then, a Finslerian dynamical model for the nonlinear complementarity problem of traffic equilibrium is presented that may be applied for different equilibrium problems.
In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. Compared with the usual way of formulating MPEC through a KKT condition, this formulation has the advantage that it does not involve extra multipliers as new variables, and it usually requires weaker assumptions on the problem data. Using the so-called first order sufficient condition for metric subregularity, we derive verifiable sufficient conditions for the metric subregularity of the involved set-…
Dynamic programming is a mathematical optimization method and a computer programming method as well. In this paper, the notion of sheaf programming in topological spaces is introduced and it is demonstrated that it relates very well to the concept of dynamic programming.
The non-convex quadratic orogramming problem and the non-monotone linear complementarity problem are NP-complete problems. In this paper we first show taht the inverse problem of determinning a KKT point of the non-convex quadratic programming problem is polynomial. We then show that the inverse problems of non-monotone linear complementarity problem are polynomial solvable in some cases, and in another case is NP-hard. Therefore we solve an open question raised by Heuberger on inverse NP-hard problems and prove that CoNP=NP.
A mathematical programming problem with affine equilibrium constraints (AMPEC) is a bilevel programming problem where the lower one is a parametric affine variational inequality. We formulate some classes of bilevel programming in forms of MPEC. Then we use a regularization technique to formulate the resulting problem as a mathematical program with an additional constraint defined by the difference of two convex functions (DC function). A main feature of this DC decomposition is that the second component depends upon only the parameter in the lower problem. This property allows us to develop b…
The paper is devoted to an analysis of a new constraint qualification and a derivation of the strongest existing optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials under the minimal possible assumptions. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints with the use of a new constraint qualification. We utilize these description and constraint qualification to derive t…
We develop and analyse a first-order algorithm for the A-optimal experimental design problem. The problem is first presented as a special case of a parametric family of optimal design problems for which duality results and optimality conditions are given. Then, two first-order (Frank-Wolfe type) algorithms are presented, accompanied by a detailed time-complexity analysis of the algorithms and computational results on various sized problems.
Several different ways exist for approaching hard optimization problems. Mathematical programming techniques, including (integer) linear programming-based methods and metaheuristic approaches, are two highly successful streams for combinatorial problems. These two have been established by different communities more or less in isolation from each other. Only over several years ago, a larger number of researchers recognized the advantages and huge potentials of building hybrids of mathematical programming methods and metaheuristics.
The dynamic time warping (dtw) distance is an established tool for mining time series data. The DTW-Mean problem consists of computing a series which minimizes the so-called Fréchet function, that is, the sum of squared dtw-distances to a given sample of time series. DTW-Mean is NP-hard and intractable in practice. So far, this challenging problem has been solved by various heuristic approaches without any performance guarantees. We give a polynomial-time algorithm yielding lower bounds on the domain of a mean time series which translate into lower bounds on the Fréchet function. We then formu…
Urban air mobility is a concept that promotes aerial modes of transport in urban areas. In these areas, the location and capacity of the vertiports--where the travelers embark and disembark the aircraft--not only affect the flight delays of the aircraft, but can also aggravate the congestion of ground vehicles by creating extra ground travel demands. We introduce a mathematical model for selecting the location and capacity of the vertiports that minimizes the traffic congestion in hybrid air-ground transportation networks. Our model is based on a mathematical program with bilinear equilibrium …
Borwein et al. (2000) solved a surprise maximization problem by applying results from convex analysis and mathematical programming. Although, their proof is elegant, it requires advanced knowledge from both areas to understand it. Here, we provide another approach to derive an optimal solution of the problem by utilizing dynamic programming.
We present ZERO, a modular and extensible C++ library interfacing Mathematical Programming and Game Theory. ZERO provides a comprehensive toolkit of modeling interfaces and algorithms for Reciprocally Bilinear Games (RBGs), i.e., simultaneous non-cooperative games where each player solves a mathematical program with a linear objective in the player's variable and bilinear in its opponents' variables. This class of games generalizes the classical problems of Operations Research to a multi-agent setting. ZERO modular structure gives users all the elementary ingredients to design new game-theoret…
We show that a direct shooting method is mathematically equivalent to an indirect method in the sense of certain first-order conditions. Specific mathematical formulas pertaining to the equivalence of a direct shooting method with an indirect method are derived. We also show that a theoretical equivalence does not necessarily translate to practical equivalence if the parameterized optimal control problem is simply patched to a nonlinear programming solver. A mathematical explanation is provided for the successes and failures of such patched nonlinear programming methods. In order to generate t…
This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we…
We present Epsilon, a system for general convex programming using fast linear and proximal operators. As with existing convex programming frameworks, users specify convex optimization problems using a natural grammar for mathematical expressions, composing functions in a way that is guaranteed to be convex by the rules of disciplined convex programming. Given such an input, the Epsilon compiler transforms the optimization problem into a mathematically equivalent form consisting only of functions with efficient proximal operators---an intermediate representation we refer to as prox-affine form.…
University of Michigan（密歇根大学）；Yasar University（亚沙尔大学）
This paper considers the integrated problem of quay crane assignment, quay crane scheduling, yard location assignment, and vehicle dispatching operations at a container terminal. The main objective is to minimize vessel turnover times and maximize the terminal throughput, which are key economic drivers in terminal operations. Due to their computational complexities, these problems are not optimized jointly in existing work. This paper revisits this limitation and proposes Mixed Integer Programming (MIP) and Constraint Programming (CP) models for the integrated problem, under some realistic ass…
We consider optimization problems with a disjunctive structure of the constraints. Prominent examples of such problems are mathematical programs with equilibrium constraints or vanishing constraints. Based on the concepts of directional subregularity and their characterization by means of objects from generalized differentiation, we obtain the new stationarity concept of extended M-stationarity, which turns out to be an equivalent dual characterization of B-stationarity. These results are valid under a very weak constraint qualification of Guignard type which is usually very difficult to verif…
This paper reviews recent advances in the field of optimization under uncertainty via a modern data lens, highlights key research challenges and promise of data-driven optimization that organically integrates machine learning and mathematical programming for decision-making under uncertainty, and identifies potential research opportunities. A brief review of classical mathematical programming techniques for hedging against uncertainty is first presented, along with their wide spectrum of applications in Process Systems Engineering. A comprehensive review and classification of the relevant publ…
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 for…
This manuscript was written because I found no other introduction to SDP programming that targets the same audience. A first difference compared to other existing introductions to SDP is that this work comes out of a mind that was itself struggling to understand. This may seem to be only a weakness, but, paradoxically, it is both a weakness and a strength. First, I did not try to overpower the reader, but I tried to minimize the distance between the author and the reader as much as possible, even hoping to achieve a small level of mutual empathy. This enabled me avoid a quite common pitfall: m…
Mathematical programs with disjunctive constraints (MPDCs for short) cover several different problem classes from nonlinear optimization including complementarity-, vanishing-, cardinality-, and switching-constrained optimization problems. In this paper, we introduce an abstract but reasonable version of the prominent linear independence constraint qualification which applies to MPDCs. Afterwards, we derive first- and second-order optimality conditions for MPDCs under validity of this constraint qualification based on so-called strongly stationary points. Finally, we apply our findings to some…
The optimal value function is one of the basic objects in the field of mathematical optimization, as it allows the evaluation of the variations in the cost/revenue generated while minimizing/maximizing a given function under some constraints. In the context of stability/sensitivity analysis, a large number of publications have been dedicated to the study of continuity and differentiability properties of the optimal value function. The differentiability aspect of works in the current literature has mostly been limited to first order analysis, with focus on estimates of its directional derivativ…
