Mixed-integer convex representability

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first non-representability results for various non-convex sets such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets

An algorithm for the global resolution of linear stochastic bilevel programs

The aim of this thesis is to find a technique that allows for the use of decomposition methods known from stochastic programming in the framework of linear stochastic bilevel problems. The uncertainty is modeled as a discrete, finite distribution on some probability space. Two approaches are made, one using the optimal value function of the lower level, whereas the second technique uses the Karush-Kuhn-Tucker conditions of the lower level. Using the latter approach, an integer-programming based algorithm for the global resolution of these problems is presented and evaluated

An accelerated Newton–Dinkelbach method and its application to two variables per inequality systems

We present an accelerated, or ‘look-ahead’ version of the Newton–Dinkelbach method, a wellknown technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains: (i) For linear fractional combinatorial optimization, we show a convergence bound of O(m log m) iterations; the previous best bound was O(m2 log m) by Wang et al. (2006). (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with n variables and m constraints, our algorithm runs in O(mn) iterations. Every iteration takes O(mn) time for general 2VPI systems, and O(m+n log n) time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (2002) that showed a weakly polynomial bound for a variant of the Newton–Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result by Goemans et al. (2017)

Calculus of unbounded spectrahedral shadows and their polyhedral approximation

The present thesis deals with the polyhedral approximation and calculus of spectrahedral shadows that are not necessarily bounded. These sets are the images of the feasible regions of semidefinite programs under linear transformations. Spectrahedral shadows contain polyhedral sets as a proper subclass. Therefore, the method of polyhedral approximation is a useful device to approximately describe them using members of the same class with a simpler structure. In the first part we develop a calculus for spectrahedral shadows. Besides showing their closedness under numerous set operations, we derive explicit descriptions of the resulting sets as spectrahedral shadows. Special attention is paid to operations that result in unbounded sets, such as the polar cone, conical hull and recession cone. The second part is dedicated to the approximation of compact spectrahedral shadows with respect to the Hausdorff distance. We present two algorithms for the computation of polyhedral approximations of such sets. Convergence as well as correctness of both algorithms are proved. As a supplementary tool we also present an algorithm that generates points from the relative interior of a spectrahedral shadow and computes its affine hull. Finally, we investigate the limits of polyhedral approximation in the Hausdorff distance in general and, extending known results, characterize the sets that admit such approximations. In the last part we develop concepts and tools for the approximation of spectrahedral shadows that are compatible with unboundedness. We present two notions of polyhedral approximation and show that sequences of approximations converge to the true set if the approximation errors diminish. In combination with algorithms for their computation we develop an algorithm for the polyhedral approximation of recession cones of spectrahedral shadows. Finiteness and correctness of all algorithms are proved and properties of the approximation concepts are investigated

A strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.Comment: minor correction

Studies integrating geometry, probability, and optimization under convexity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2006.Includes bibliographical references (p. 197-202).Convexity has played a major role in a variety of fields over the past decades. Nevertheless, the convexity assumption continues to reveal new theoretical paradigms and applications. This dissertation explores convexity in the intersection of three fields, namely, geometry, probability, and optimization. We study in depth a variety of geometric quantities. These quantities are used to describe the behavior of different algorithms. In addition, we investigate how to algorithmically manipulate these geometric quantities. This leads to algorithms capable of transforming ill-behaved instances into well-behaved ones. In particular, we provide probabilistic methods that carry out such task efficiently by exploiting the geometry of the problem. More specific contributions of this dissertation are as follows. (i) We conduct a broad exploration of the symmetry function of convex sets and propose efficient methods for its computation in the polyhedral case. (ii) We also relate the symmetry function with the computational complexity of an interior-point method to solve a homogeneous conic system. (iii) Moreover, we develop a family of pre-conditioners based on the symmetry function and projective transformations for such interior-point method.(cont.) The implementation of the pre-conditioners relies on geometric random walks. (iv) We developed the analysis of the re-scaled perceptron algorithm for a linear conic system. In this method a sequence of linear transformations is used to increase a condition measure associated with the problem. (v) Finally, we establish properties relating a probability density induced by an arbitrary norm and the geometry of its support. This is used to construct an efficient simulating annealing algorithm to test whether a convex set is bounded, where the set is represented only by a membership oracle.by Alexandre Belloni Nogueira.Ph.D

On orbital allotments for geostationary satellites

The following satellite synthesis problem is addressed: communication satellites are to be allotted positions on the geostationary arc so that interference does not exceed a given acceptable level by enforcing conservative pairwise satellite separation. A desired location is specified for each satellite, and the objective is to minimize the sum of the deviations between the satellites' prescribed and desired locations. Two mixed integer programming models for the satellite synthesis problem are presented. Four solution strategies, branch-and-bound, Benders' decomposition, linear programming with restricted basis entry, and a switching heuristic, are used to find solutions to example synthesis problems. Computational results indicate the switching algorithm yields solutions of good quality in reasonable execution times when compared to the other solution methods. It is demonstrated that the switching algorithm can be applied to synthesis problems with the objective of minimizing the largest deviation between a prescribed location and the corresponding desired location. Furthermore, it is shown that the switching heuristic can use no conservative, location-dependent satellite separations in order to satisfy interference criteria