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.............................................................................................................................. ii Co-Authorship Statement................................................................................................... iv Epigraph .............................................................................................................................. v Ded...

An algorithm is suggested for solving a convex programming problem which belongs to a class of cutting methods. In the algorithm an epigraph of the objective function and a feasible solutions set of the problem are embedded into some auxiliary sets to construct iteration points. Since these embedded sets are constructed as polyhedral sets in the algorithm, then each iteration point is found by ...

This paper concerns rigidity results to Serrin’s overdetermined problem in an epigraph  ∆u + f(u) = 0, in Ω = {(x′, xn) : xn > φ(x′)}, u > 0, in Ω, u = 0, on ∂Ω, |∇u| = const., on ∂Ω. We prove that up to isometry the epigraph must be an half space and that the solution u must be one-dimensional, provided that one of the following assumptions are satisfied: either n = 2; or φ is globally ...

We propose and study a new method, called the Interior Epigraph Directions (IED) method, for solving constrained nonsmooth and nonconvex optimization. The IED method considers the dual problem induced by a generalized augmented Lagrangian duality scheme, and obtains the primal solution by generating a sequence of iterates in the interior of the dual epigraph. First, a deflected subgradient (DSG...

2010 iii DEDICATION To all those willing to wait for the fruits of their labor. iv EPIGRAPH When the facts change, I change my mind. What do you do, sir?

We define the notions of lower and upper semicontinuity for functions from a metric space to the extended real line. We prove that a function is both lower and upper semicontinuous if and only if it is continuous. We also give several equivalent characterizations of lower semicontinuity. In particular, we prove that a function is lower semicontinuous if and only if its epigraph is a closed set....

By using properties of dualizing parametrization functions, Lagrangian functions and the epigraph technique, some sufficient and necessary conditions of the stable strong duality and strong total duality for a class of DC optimization problems are established.

We study the trust region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone based approach to derive an exact convex formulation of the TRS, and under slightly stro...