A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York

A reducibility method for the weak linear bilevel programming problems and a case study in principal-agent

© 2018 A weak linear bilevel programming (WLBP) problem often models problems involving hierarchy structure in expert and intelligent systems under the pessimistic point. In the paper, we deal with such a problem. Using the duality theory of linear programming, the WLBP problem is first equivalently transformed into a jointly constrained bilinear programming problem. Then, we show that the resolution of the jointly constrained bilinear programming problem is equivalent to the resolution of a disjoint bilinear programming problem under appropriate assumptions. This may give a possibility to solve the WLBP problem via a single-level disjoint bilinear programming problem. Furthermore, some examples illustrate the solution process and feasibility of the proposed method. Finally, the WLBP problem models a principal-agent problem under the pessimistic point that is also compared with a principal-agent problem under the optimistic point

Topics in Mixed Integer Nonlinear Optimization

Mixed integer nonlinear optimization has many applications ranging from machine learning to power systems. However, these problems are very challenging to solve to global optimality due to the inherent non-convexity. This typically leads the problem to be NP-hard. Moreover, in many applications, there are time and resource limitations for solving real-world problems, and the sheer size of real instances can make solving them challenging. In this thesis, we focus on important elements of nonconvex optimization - including mixed integer linear programming and nonlinear programming, where both theoretical analyses and computational experiments are presented. In the first chapter we look at Mixed Integer Quadratic Programming (MIQP), the problem of minimizing a convex quadratic function over mixed integer points in a rational polyhedron. We utilize the augmented Lagrangian dual (ALD), which augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function. Finally, we prove a polynomial bound on the weight on the penalty term to obtain a zero gap. In the second chapter we apply the technique of lifting to bilinear programming, a special case of quadratic constrained quadratic programming. We first show that, for sets described by one bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality. To reduce computational burden, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bilinear sets. We then study a separable bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We study the lifting function of the bilinear cover inequality and lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bilinear sets with box constraints. In the third chapter we continue our research around separable bilinear programming. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for separable bilinear optimization problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from the McCormick relaxation of bilinear instances where each constraint is a separable bilinear constraint set. Our main result is to demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of the gap closed, when these inequalities are added at the root node compared to when these inequalities are not added. In the fourth chapter we look at Mixed Integer Linear Programming (MILP) that arises in operational applications. Many routinely-solved MILPs are extremely challenging not only from a worst-case complexity perspective, but also because of the necessity to obtain good solutions within limited time. An example is the Security-Constrained Unit Commitment (SCUC) problem, solved daily to clear the day-ahead electricity markets. We develop ML-based methods for improving branch-and-bound variable selection rules that exploit key features of such operational problems: similar decisions are generated within the same day and across different days, based on the same power network. Utilizing similarity between instances and within an instance, we build one separate ML model per variable or per group of similar variables for learning to predict the strong branching score. The approach is able to produce branch-and-bound trees which gap closed only slightly worse than that of trees obtained by strong branching, while it outperforms previous machine learning schemes.Ph.D

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

(Global) Optimization: Historical notes and recent developments

Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning

Déploiement et mise à jour de coupes de concavité

RÉSUMÉ : Dans le cadre de minimisation concave, un type de problème d'optimisation quadratique, nommé bilinéaire disjoint (BILD), peut se reformuler en deux problèmes linéaires symétriques MinMax (LMM). Comme les solutions optimales de BILD et de ses reformulations LMM sont liées par une simple bijection, la question de notre travail de recherche s'agit profiter de cette reformulation pour résoudre BILD. Dans la littérature, une technique de calcul, appelée coupe de concavitée, proposée par Tuy [39] en 1964, s'avère importante afin de résoudre les BILD. L'algorithme basé uniquement sur cette technique n'est pas sûrement de convergence finie. Cela est dû à plusieurs problèmes, parmi lesquels nous citons : le problème de dégénérescene et le problème de cumul des coupes. Depuis, des chercheurs tentent d'améliorer la convergence de l'algorithme des plans coupants. Pour ce faire, ils ont intégré cet algorithme avec d'autres techniques de calcul. En effet, Konno [23] a introduit la technique Mountain Climbing en 1971 pour évaluer à chaque itération une solution locale. Avec l'usage des pseudo-sommets, Marcus [34] a developpé en 1999 des coupes similaires à celles de Tuy en décomposant le cône polyèdral. En 2001, Alarie et al. ont traité le problème de dégénérescence et ont exploité la technique de "Branch and Bound" pour les instances BILD de grande dimension. La recherche proposée dans ce mémoire se situe dans la continuation de ces travaux. Nous proposons deux nouvelles stratégies pour la génération de coupes de concavité. Dans la première, nous avons élaboré une mise à jour dynamique des coupes après qu'une améioration de la valeur objectif soit faite. Dans la seconde, au lieu que les coupes soient associées à des sommets, nous les avons associées aux pseudo-sommets. Ces deux nouvelles stratégies sont testées numériquement sur un ensemble de problèmes tests tirés de la littérature ainsi que sur une collection de problèmes générés aléatoirement.----------ABSTRACT : In the context of concave minimisation, a type of quadratic optimization problem, called BILD problem, can be reformulated into two symmetrical linear MinMax problems LMM. As there is a simple bijection between the optimal solution of BILD and their reformulations LMM, our research question is to take advantage of this reformulations for solving BILD. In the litterature, a computation technique , called concavity cut, proposed by Tuy [39] in 1964, has been shown to be useful solving the BILD problem. However, it is still unknown whether the nite convergence of a cutting planes algorithm can be enforced by the concavity cut itself or not. This is due to several problems, among which we mention : the degeneracy problem and the accumulation of cuts. Since then, researchers have attempted to improve the convergence of the cutting planes algorithm. To achieve this, they have integrated the algorithm with other computation techniques. Indeed, Konno [23] introduced in 1971 the mountain climbing (MC) technique to evaluate in each iteration a local optimal solution. In 1999, Marcus [34] used the pseudo-vertices to developed similar Tuy cuts by decomposing the polyhedral cone. Alarie and al. treated in 2001 the degeneracy problem ant they have exploited the "Branch and Bound" technique for solvign BILD instances with large dimensions. The proposed research in this project is the continuation of this works. Thus, we proposed two ways for deploying cuts. In the rst one, we dynamically update previously generated cuts after an improvement of the objective function value. In the second, instead rooting the cuts at vertices, we root them at pseudo-vertices. These two new strategies are tested numerically on a set of test problems issued from the literature as well as on a collection of randomly generated instances

Global Optimisation for Energy System

The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity