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.
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.
Inverse linear programming (LP) has received increasing attention due to its potential to generate efficient optimization formulations that can closely replicate the behavior of a complex system. However, inversely inferred parameters and corresponding forward solutions from the existing inverse LP method can be highly sensitive to noise, errors, and uncertainty in the input data, limiting its applicability in data-driven settings. We introduce the notion of inverse and forward stability in inverse LP and propose a novel inverse LP method that determines a set of objective functions that are s…
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 …
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 …
Disaster management is a complex problem demanding sophisticated modeling approaches. We propose utilizing a hybrid method involving inverse optimization to parameterize the cost functions for a road network's traffic equilibrium problem and employing a modified version of Fan and Liu (2010)'s two-stage stochastic model to make protection decisions using the information gained from inverse optimization. We demonstrate the framework using two types of cost functions for the traffic equilibrium problem and show that accurate parameterizations of cost functions can change spending on protection d…
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…
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…
In this work we study a constrained monotone inclusion involving the normal cone to a closed vector subspace and a priori information on primal solutions. We model this information by imposing that solutions belongs to the fixed point set of an averaged nonexpansive mapping. We characterize the solutions using an auxiliary inclusion that involves the partial inverse operator. Then, we propose the primal-dual partial inverse splitting and we prove its weak convergence to a solution of the inclusion, generalizing several methods in the literature. The efficiency of the proposed method is illustr…
Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenge…
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…
A systematic literature review on the application of analytical approaches and mathematical programming in public bus transit network design and operations planning: Part I
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 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…
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.
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 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…
In this work we consider algorithms for reconstructing time-varying data into a finite sum of discrete trajectories, alternatively, an off-the-grid sparse-spikes decomposition which is continuous in time. Recent work showed that this decomposition was possible by minimising a convex variational model which combined a quadratic data fidelity with dynamical Optimal Transport. We generalise this framework and propose new numerical methods which leverage efficient classical algorithms for computing shortest paths on directed acyclic graphs. Our theoretical analysis confirms that these methods conv…
