In this paper, we present a semi-proximal alternating direction method of multipliers (ADMM) for solving 3-block separable convex minimization problems with the second block in the objective being a strongly convex function and one coupled linear equation constraint. By choosing the semi-proximal terms properly, we establish the global convergence of the proposed semi-proximal ADMM for the step...

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set K ⊂ Rn. Namely, we prove that almost all of them belong to a specific subset of the quadratic module generated by the concave polynomials that define K. In particular, almost all nonnegative convex polynomials are sums of squares. Mathematics Subject Class...

In this paper, we aim to provide a comprehensive analysis on the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a certain error bound condition, we establish the global linear rate of convergence for a more general semi-proximal ADMM with the dual steplength being restricted to be i...

Generalized semi-infinite optimization problems (GSIP) are considered. It is investigated how the numerical methods for standard semi-infinite programming (SIP) can be extended to GSIP. Newton methods can be extended immediately. For discretization methods the situation is more complicated. These difficulties are discussed and convergence results for a discretizationand an exchange method are d...

In this paper, we introduce and study a class of generalized symmetric vector quasi-equilibrium problems in abstract convex spaces. By virtue of the properties of Γ-convex and KC-map, we give some sufficient conditions to guarantee the existence of solutions for the generalized symmetric vector quasi-equilibrium problems in abstract convex spaces. As application, we show an existence theorem of...

| We consider the problem of synthesizing feasible signals in a Hilbert space in the presence of inconsistent convex constraints, some of which must imperatively be satissed. This problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones. The resulting convex optimizati...

We prove that for certain families of semi-algebraic convex bodies in R3, the convex hull of n disjoint bodies has O(nλs(n)) features, where s is a constant depending on the family: λs(n) is the maximum length of order-s Davenport-Schinzel sequences with n letters. The argument is based on an apparently new idea of ‘compact families’ of convex bodies or discs, and of ‘crossing content’ and ‘foo...

This is a set of notes that is basically and expanded version of the paper Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls. The differences are a few extra pictures, Section 2.7 which is an exposition of results of Ng and Nikodem [5] about measurable approximately convex functions, and an alternate proof of Theorem 2.27 is included. Contents 1. Introduction 2 2. ...