In this paper we consider the linear symmetric cone programming (SCP). At a KarushKuhn-Tucker (KKT) point of SCP, we present the important equivalent conditions for the nonsingularity of Clarke’s generalized Jacobian of the KKT nonsmooth system, such as primal and dual constraint nondegeneracy, the strong regularity, and the nonsingularity of the B-subdifferential of the KKT system. This affirm...

We present a strong duality theory for optimization problems over symmetric cones without assuming any constraint qualification. We show important complexity implications of the result to semidefinite and second order conic optimization. The result is an application of Borwein and Wolkowicz’s facial reduction procedure to express the minimal cone. We use Pataki’s simplified analysis and provide...

We consider a chance constraint Prob{ξ : A(x, ξ) ∈ K} ≥ 1 − 2 (x is the decision vector, ξ is a random perturbation, K is a closed convex cone, and A(·, ·) is bilinear). While important for many applications in Optimization and Control, chance constraints typically are “computationally intractable”, which makes it necessary to look for their tractable approximations. We present these approximat...

It is well known that the robust counterpart introduced by Ben-Tal and Nemirovski [2] increases the numerical complexity of the solution compared to the original problem. Kočvara, Nemirovski and Zowe therefore introduced in [9] an approximation algorithm for the special case of robust material optimization, called cascading. As the title already indicates, we will show that their method can be ...

In this paper we consider a minimum distance Controlled Tabular Adjustment (CTA) model for statistical disclosure limitation (control) of tabular data. The goal of the CTA model is to find the closest safe table to some original tabular data set that contains sensitive information. The measure of closeness is usually measured using `1 or `2 norm; with each measure having its advantages and disa...

A Standard Quadratic Optimization Problem (StQP) consists of maximizing a (possibly indefinite) quadratic form over the standard simplex. Likewise, in a multi-StQP we have to maximize a (possibly indefinite) quadratic form over the cartesian product of several standard simplices (of possibly different dimensions). Two converging monotone interior point methods are established. Further, we prove...

For a conic optimization problem P : minimizex c x s.t. Ax = b, x ∈ C and its dual D : supremumy,s b T y s.t. A y + s = c, s ∈ C, we present a geometric relationship between the primal objective function level sets and the dual objective function level sets, which shows that the maximum norms of the primal objective function level sets are nearly inversely proportional to the maximum inscribed ...

An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositiv...

A homogeneous infeasible-start interior-point algorithm for solving nonsymmetric convex conic optimization problems is presented. Starting each iteration from the vicinity of the central path, the method steps in the approximate tangent direction and then applies a correction phase to locate the next well-centered primal-dual point. Features of the algorithm include that it makes use only of th...

A “facial reduction”-like regularization algorithm is established for general conic optimization problems by relaxing requirements on the reduction certificates. This yields a rapid subspace reduction algorithm challenged only by representational issues of the regularized cone. A condition for practical usage is analyzed and shown to always be satisfied for single second-order cone optimization...