Abstract. We consider a conic-quadratic (and in particular a quadratically constrained) optimization problem with uncertain data, known only to reside in some uncertainty set U . The robust counterpart of such a problem leads usually to an NP-hard semidefinite problem; this is the case, for example, when U is given as the intersection of ellipsoids or as an n-dimensional box. For these cases we...

The robustification of trading strategies is of particular interest in financial market applications. In this paper we robustify a portfolio strategy recently introduced in the literature against model errors in the sense of a worst case design. As it turns out, the resulting optimization problem can be solved by a sequence of linear and nonlinear semidefinite programs (SDP/NSDP), where the non...

Let C be a real-valued function defined on the set 9& of all positive definite complex hermitian or real symmetric matrices according as F = C (the complex field) or F = R (the real field). Suppose A, B E 9&. We study the optimization problems of (1) finding max C(X) subject to A X, B X positive semidefinite, (2) finding minG(X) subject to X A, X B positive semidefinite. For a general class of ...

Many combinatorial optimization problems have relaxations that are semidefinite programming problems. In principle, the combinatorial optimization problem can then be solved by using a branch-and-cut procedure, where the problems to be solved at the nodes of the tree are semidefinite programs. It is desirable that the solution to one node of the tree should be exploited at the child node in ord...

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in ...

Given a finite set of outlets with joint normally distributed demands and identical holding and penalty costs, inventory centralization induces a cooperative cost allocation game with nonempty core. It is well known that for this newsvendor inventory setting the expected cost of centralization can be expressed as a constant multiple of the standard deviation of the joint distribution. The lower...

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear nonlinear systems theory. However, the scale of problems that can be addressed reliably efficiently is still limited. In this paper, we introduce a new notion block factor-width-two matrices build hierarchy inner outer approximations cone positive semidefinite (PSD) matrices. T...

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimizati...

Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in Hk(Ω), or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in maki...