Applications of accuracy certificates for problems with convex structure

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Cox, Bruce
Nemirovski, Arkadi
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Applications of accuracy certificates for problems with convex structure   This dissertation addresses the efficient generation and potential applications of accuracy certificates in the framework of “black-box-represented” convex optimization problems - convex problems where the objective and the constraints are represented by  “black boxes” which, given on input a value x of the argument, somehow (perhaps in a fashion unknown to the user) provide on output the values and the derivatives of the objective and the constraints at x. The main body of the dissertation can be split into three parts.  In the first part, we provide our background --- state of the art of the theory of accuracy certificates for black-box-represented convex optimization. In the second part, we extend the toolbox of black-box-oriented convex optimization algorithms with accuracy certificates by equipping with these certificates a state-of-the-art algorithm for large-scale nonsmooth black-box-represented problems with convex structure, specifically, the Non-Euclidean Restricted Memory Level (NERML) method. In the third part, we present several novel academic applications of accuracy certificates. The dissertation is organized as follows: In Chapter 1, we motivate our research goals and present a detailed summary of our results. In Chapter 2, we outline the relevant background, specifically, describe four generic black-box-represented generic problems with convex structure (Convex Minimization, Convex-Concave Saddle Point, Convex Nash Equilibrium, and Variational Inequality with Monotone Operator), and outline the existing theory of accuracy certificates for these problems. In Chapter 3, we develop techniques for equipping with on-line accuracy certificates the state-of-the-art NERML algorithm for large-scale nonsmooth problems with convex structure, both in the cases when the domain of the problem is a simple solid and in the case when the domain is given by Separation oracle. In Chapter 4, we develop  several novel academic applications of accuracy certificates, primarily to (a) efficient certifying emptiness of the intersection of finitely many solids given by Separation oracles, and (b) building efficient algorithms for convex minimization over solids given by Linear Optimization oracles (both precise and approximate). In Chapter 5, we apply accuracy certificates to efficient decomposition of “well structured” convex-concave saddle point problems, with applications to computationally attractive decomposition of a large-scale LP program with the constraint matrix which becomes block-diagonal after eliminating a relatively small number of possibly dense columns (corresponding to “linking variables”) and possibly dense rows (corresponding to “linking constraints”).  
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