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In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry and linear algebra as well. These were active research areas, but as there was little(More)
Detecting infeasibility in conic optimization and providing certificates for infeasibility pose a bigger challenge than in the linear case due to the lack of strong duality. In this paper we generalize the approximate Farkas lemma of Todd and Ye [12] from the linear to the general conic setting, and use it to propose stopping criteria for interior point(More)
We study the convex hull of the intersection of a convex set E and a disjunctive set. This intersection is at the core of solution techniques for Mixed Integer Convex Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction as E , then the convex hull is the intersection(More)
In this paper we investigate families of quadrics that have fixed intersections with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are transformed as the parameter changes. This(More)
In this paper we introduce a novel reinforcement learning algorithm called event-learning. The algorithm uses events, ordered pairs of two consecutive states. We define event-value function and we derive learning rules. Combining our method with a well-known robust control method, the SDS algorithm, we introduce Robust Policy Heuristics (RPH). It is shown(More)
We investigate the derivation of disjunctive conic cuts for mixed integer second order cone optimization (MISOCO). These conic cuts characterize the convex hull of the intersection of a disjunctive set and the feasible set of a MISOCO problem. We present a full characterization of these inequalities when the disjunctive set considered is defined by parallel(More)
We present two novel applications of symmetries for mixedinteger linear programming. First we propose two variants of a new heuristic to improve the objective value of a feasible solution using symmetries. These heuristics can use either the actual permutations or the orbits of the variables to find better feasible solutions. Then we introduce a new class(More)
We give an overview of cone optimization software, with special attention to the differences between the existing packages. We assume the reader is familiar with the theory and algorithms of cone optimization, thus technical details are kept at a minimum. We also outline current research trends and area of potential improvement. 1 Problem description Conic(More)