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

Perturbed cones for analysis of uncertain multi-criteria optimization problems

AbstractPartial ordering of two quantities x and y (i.e., the ability to declare that x is better than y with respect to some decision criteria) can be stated mathematically as: x is better than y iff x−y∈K, where K is an ordering convex cone, not necessarily pointed. Cones can be very important in representing feasible domains (i.e., {Ax⩽b}=M+G, where M is a bounded convex hull of a finite number of points and G is a convex cone). We consider specific perturbations of the Cone of Feasible Directions, which lead to a better feasible solution with respect to some decision criteria. Such cones are introduced as a tool to mitigate and analyze the effects of input data uncertainty on the solution of a given problem. Properties of this cone provide a basis to prove necessary and sufficient conditions for stable/unstable unboundedness of the multi-criteria optimization problem

Convex Constrained Programmes with Unattained Infima

AbstractWe consider a problem of minimization of convex function f(x) over the convex region R where the objective function and the feasible region have a common direction of recession. In cases when one of these directions is not in the constancy space of the objective function, then the minimal solution is not achieved even if the function f(x) is bounded below over the region R. Many algorithms, if applied to this class of programmes, do not guarantee convergence to the global infimum. Our approach to this problem leads to derivation of the equation of the feasible parametrized curve C(t), such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region R. We show that if all functions defining the program are analytic, then C(t) is also an analytic function. The equation of the curve can be successfully used to determine the global infimum (in particular, unboundedness) of the convex constrained programmes in cases when the application of classical methods, such as the steepest descent method, fails to converge to the global infimum

Asymptotic Cones of Quadratically Defined Sets and Their Applications to QCQPs

Quadratically constrained quadratic programs (QCQPs) are a set of optimization problems defined by a quadratic objective function and quadratic constraints. QCQPs cover a diverse set of problems, but the nonconvexity and unboundedness of quadratic constraints lead to difficulties in globally solving a QCQP. This thesis covers properties of unbounded quadratic constraints via a description of the asymptotic cone of a set defined by a single quadratic constraint. A description of the asymptotic cone is provided, including properties such as retractiveness and horizon directions. Using the characterization of the asymptotic cone, we generalize existing results for bounded quadratically defined regions with non-intersecting constraints. The newer result provides a sufficient condition for when the intersection of the lifted convex hulls of quadratically defined sets equals the lifted convex hull of the intersection. This document goes further by expanding the non-intersecting property to cover affine linear constraints. The Frank-Wolfe theorem provides conditions for when a problem defined by a quadratic objective function over affine linear constraints has an optimal solution. Over time, this theorem has been extended to cover cases involving convex quadratic constraints. We discuss more current results through the lens of the asymptotic cone of a quadratically defined set. This discussion expands current results and provides a sufficient condition for when a QCQP with one quadratic constraint with an indefinite Hessian has an optimal solution

Existence of Equilibrium in Incomplete Markets with Non-Ordered Preferences

In this paper we extend the results of recent studies on the existence of equilibrium in finite dimensional asset markets for both bounded and unbounded economies. We do not assume that the individual's preferences are complete or transitive. Our existence theorems for asset markets allow for short selling. We shall also show that the equilibrium achieves a constrained core within the same framework.Equilibrium Existence, Incomplete Preferences, Incomplete Markets, Constrained Core

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