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Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables.…

In this paper, in terms of three types of generalized second-order derivatives of a nonsmooth function, we mainly study the corresponding second-order optimality conditions in a Hilbert space and prove the equivalence among these optimality…

Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point…

We prove some fundamental properties of mu-differentiable functions. A new notion of local minimizer and maximizer is introduced and several extremum conditions are formulated using the language of nonstandard analysis.

A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an…

In applied mathematics, especially in optimization, functions are often only provided as so called "Black-Boxes" provided by software packages, or very complex algorithms, which make automatic differentation very complicated or even…

Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science.…

This paper identifies necessary and sufficient conditions for the exactness of penalty functions in optimization problems whose constraint sets are not necessarily bounded. The case where the data of problems is locally Lipschitz,…

In this paper, we readdress the classical topic of second-order sufficient optimality conditions for optimization problems with nonsmooth structure. Based on the so-called second subderivative of the objective function and of the indicator…

Many protocols of quantum information processing use entangled states. Hence, separability criteria are of great importance. We propose new separability conditions for a bipartite finite-dimensional system. They are derived by using…

In the past decade, we had developed a series of splitting contraction algorithms for separable convex optimization problems, at the root of the alternating direction method of multipliers. Convergence of these algorithms was studied under…

We propose a novel distribution-free scheme to solve optimization problems where the goal is to minimize the expected value of a cost function subject to probabilistic constraints. Unlike standard sampling-based methods, our idea consists…

Optimization problems with both control variables and environmental variables arise in many fields. This paper introduces a framework of personalized optimization to han- dle such problems. Unlike traditional robust optimization,…

Submodularity is one of the most well-studied properties of problem classes in combinatorial optimization and many applications of machine learning and data mining, with strong implications for guaranteed optimization. In this thesis, we…

This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem…

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel…

Since more than three decades, interior-point methods proved very useful for optimization, from linear over semidefinite to conic (and partly beyond non-convex) programming; despite the fact that already in the semidefinite case (even when…

Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct randomized algorithm for solving large, weakly constrained SDP…

Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical…

In this paper we consider distributed optimization problems in which the cost function is separable, i.e., a sum of possibly non-smooth functions all sharing a common variable, and can be split into a strongly convex term and a convex one.…