Combinatorial Optimization

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry

Intersection disjunctions for reverse convex sets

We present a framework to obtain valid inequalities for optimization problems constrained by a reverse convex set, which is defined as the set of points in a polyhedron that lie outside a given open convex set. We are particularly interested in cases where the closure of the convex set is either non-polyhedral, or is defined by too many inequalities to directly apply disjunctive programming. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. Intersection cuts are a well-known method for generating valid inequalities for a reverse convex set. Intersection cuts are generated from a basic solution that lies within the convex set. Our contribution is a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We begin by proposing an extension to intersection cuts that defines a two-term disjunction for a reverse convex set. Next, we generalize this analysis to a multi-term disjunction by considering the convex set's recession directions. These disjunctions can be used in a cut-generating linear program to obtain disjunctive cuts for the reverse convex set.Comment: 24 page

Efficient Algorithms for Solving Facility Problems with Disruptions

This study investigates facility location problems in the presence of facility disruptions. Two types of problems are investigated. Firstly, we study a facility location problem considering random disruptions. Secondly, we study a facility fortification problem considering disruptions caused by random failures and intelligent attacks.We first study a reliable facility location problem in which facilities are faced with the risk of random disruptions. In the literature, reliable facility location models and solution methods have been proposed under different assumptions of the disruption distribution. In most of these models, the disruption distribution is assumed to be completely known, that is, the disruptions are known to be uncorrelated or to follow a certain distribution. In practice, we may have only limited information about the distribution. In this work, we propose a robust reliable facility location model that considers the worst-case distribution with incomplete information. Because the model imposes fewer distributional assumptions, it includes several important reliable facility location problems as special cases. We propose an effective cutting plane algorithm based on the supermodularity of the problem. For the case in which the distribution is completely known, we develop a heuristic algorithm called multi-start tabu search to solve very large instances.In the second part of the work, we study an r-interdiction median problem with fortification that simultaneously considers two types of disruption risks: random disruptions that happen probabilistically and disruptions caused by intentional attacks. The problem is to determine the allocation of limited facility fortification resources to an existing network. The problem is modeled as a bi-level programming model that generalizes the r-interdiction median problem with probabilistic fortification. The lower level problem, that is, the interdiction problem, is a challenging high-degree non-linear model. In the literature, only the enumeration method is applied to solve a special case of the problem. By exploring the special structure property of the problem, we propose an exact cutting plane method for the problem. For the fortification problem, an effective logic based Benders decomposition algorithm is proposed

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Global optimization algorithms for semi-infinite and generalized semi-infinite programs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008.Includes bibliographical references (p. 235-249).The goals of this thesis are the development of global optimization algorithms for semi-infinite and generalized semi-infinite programs and the application of these algorithms to kinetic model reduction. The outstanding issue with semi-infinite programming (SIP) was a methodology that could provide a certificate of global optimality on finite termination for SIP with nonconvex functions participating. We have developed the first methodology that can generate guaranteed feasible points for SIP and provide e-global optimality on finite termination. The algorithm has been implemented in a branch-and-bound (B&B) framework and uses discretization coupled with convexification for the lower bounding problem and the interval constrained reformulation for the upper bounding problem. Within the framework of SIP we have also proposed a number of feasible-point methods that all rely on the same basic principle; the relaxation of the lower-level problem causes a restriction of the outer problem and vice versa. All these methodologies were tested using the Watson test set. It was concluded that the concave overestimation of the SIP constraint using McCormcick relaxations and a KKT treatment of the resulting expression is the most computationally expensive method but provides tighter bounds than the interval constrained reformulation or a concave overestimator of the SIP constraint followed by linearization. All methods can work very efficiently for small problems (1-3 parameters) but suffer from the drawback that in order to converge to the global solution value the parameter set needs to subdivided. Therefore, for problems with more than 4 parameters, intractable subproblems arise very high in the B&B tree and render global solution of the whole problem infeasible.(cont.) The second contribution of the thesis was the development of the first finite procedure that generates guaranteed feasible points and a certificate of e-global optimality for generalized semi-infinite programs (GSIP) with nonconvex functions participating. The algorithm employs interval extensions on the lower-level inequality constraints and then uses discretization and the interval constrained reformulation for the lower and upper bounding subproblems, respectively. We have demonstrated that our method can handle the irregular behavior of GSIP, such as the non-closedness of the feasible set, the existence of re-entrant corner points, the infimum not being attained and above all, problems with nonconvex functions participating. Finally, we have proposed an extensive test set consisting of both literature an original examples. Similar to the case of SIP, to guarantee e-convergence the parameter set needs to be subdivided and therefore, only small examples (1-3 parameters) can be handled in this framework in reasonable computational times (at present). The final contribution of the thesis was the development of techniques to provide optimal ranges of valid reduction between full and reduced kinetic models. First of all, we demonstrated that kinetic model reduction is a design centering problem and explored alternative optimization formulations such as SIP, GSIP and bilevel programming. Secondly, we showed that our SIP and GSIP techniques are probably not capable of handling large-scale systems, even if kinetic model reduction has a very special structure, because of the need for subdivision which leads to an explosion in the number of constraints. Finally, we propose alternative ways of estimating feasible regions of valid reduction using interval theory, critical points and line minimization.by Panayiotis Lemonidis.Ph.D

Online Subset Selection using α\alpha-Core with no Augmented Regret

We consider the problem of sequential sparse subset selections in an online learning setup. Assume that the set $[N]$ consists of $N$ distinct elements. On the $t^{\text{th}}$ round, a monotone reward function $f_t: 2^{[N]} \to \mathbb{R}_+,$ which assigns a non-negative reward to each subset of $[N],$ is revealed to a learner. The learner selects (perhaps randomly) a subset $S_t \subseteq [N]$ of $k$ elements before the reward function $f_t$ for that round is revealed $(k \leq N)$. As a consequence of its choice, the learner receives a reward of $f_t(S_t)$ on the $t^{\text{th}}$ round. The learner's goal is to design an online subset selection policy to maximize its expected cumulative reward accrued over a given time horizon. In this connection, we propose an online learning policy called SCore (Subset Selection with Core) that solves the problem for a large class of reward functions. The proposed SCore policy is based on a new concept of $\alpha$-Core, which is a generalization of the notion of Core from the cooperative game theory literature. We establish a learning guarantee for the SCore policy in terms of a new performance metric called $\alpha$-augmented regret. In this new metric, the power of the offline benchmark is suitably augmented compared to the online policy. We give several illustrative examples to show that a broad class of reward functions, including submodular, can be efficiently learned with the SCore policy. We also outline how the SCore policy can be used under a semi-bandit feedback model and conclude the paper with a number of open problems

Selection methods for subgame perfect Nash equilibrium in a continuous setting

This thesis is focused on the issue of selection of Subgame Perfect Nash Equilibrium (SPNE) in the class of one-leader N-follower two-stage games where the players have a continuum of actions. We are mainly interested in selection methods satisfying the following significant features in the theory of equilibrium selection for such a class of games: obtaining an equilibrium selection by means of a constructive (in the sense of algorithmic) and motivated procedure, overcoming the difficulties due to the possible non-single-valuedness of the followers' best reply correspondence, providing motivations that would induce players to choose the actions leading to the designed selection, and revealing the leader to know the followers' best reply correspondence. Firstly, we analyze the case where the followers' best reply correspondence is assumed to be single-valued: in this case we show that finding SPNEs is equivalent to find the Stackelberg solutions of the Stackelberg problem associated to the game. Moreover, as regards to the related arising issue of the sufficient conditions ensuring the uniqueness of the followers' best reaction, we prove an existence and uniqueness result for Nash equilibria in two-player normal-form games where the action sets are Hilbert spaces and which allows the two compositions of the best reply functions to be not a contraction mapping. Furthermore, by applying such a result to the class of weighted potential games, we show the (lack of) connections between the Nash equilibria and the maximizers of the potential function. Then, in the case where the followers' best reply correspondence is not assumed to be single-valued, we examine preliminarily the SPNE selections deriving by exploiting the solutions of broadly studied problems in Optimization Theory (like the strong Stackelberg, the weak Stackelberg and the intermediate Stackelberg problems associated to the game). Since such selection methods, although behaviourally motivated, do not fit all the desirable features mentioned before, we focus on designing constructive methods to select an SPNE based on the Tikhonov regularization and on the proximal point methods (linked to the Moreau-Yosida regularization). After illustrated these two tools both in the optimization framework and in the applications to the selection of Nash equilibria in normal-form games, we present a constructive selection method for SPNEs based on the Tikhonov regularization in one-leader N-follower two-stage games (with N=1 and N>1), and a constructive selection method for SPNEs based on a learning approach which has a behavioural interpretation linked to the costs that players face when they deviate from their current actions (relying on the proximal point methods) in one-leader one-follower two-stage games