This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transforma...

We call a property generic if it holds for almost all problem instances. For linear conic problems, it has been shown in the literature that properties like uniqueness, strict complementarity or nondegeneracy of the optimal solution are generic under the assumption that Slater’s condition is fulfilled. The possibility that Slater’s condition generically fails has not been excluded. In this pape...

In the study of interior-point methods for nonsymmetric conic optimization and their applications, Nesterov [5] introduced the power cone, together with a 4-selfconcordant barrier for it. In his PhD thesis, Chares [2] found an improved 3-selfconcordant barrier for the power cone. In addition, he introduced the generalized power cone, and conjectured a “nearly optimal” self-concordant barrier fo...

In this paper, we consider both a variant of Tseng’s modified forward-backward splitting method and an extension of Korpelevich’s method for solving hemivariational inequalities with Lipschitz continuous operators. By showing that these methods are special cases of the hybrid proximal extragradient method introduced by Solodov and Svaiter, we derive iteration-complexity bounds for them to obtai...

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A new scheme to cope with two-stage stochastic optimization problems uses a risk measure as the objective function of the recourse action, where the risk measure is defined as the worst-case expected values over a set of constrained distributions. This paper develops an approach to deal with the case where both the first and second stage objective functions are convex linearquadratic. It is sho...