Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials via the definition of convexity, its first order characterization, and its second order characterization are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming … Read more
In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or ‘lift’ of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over … Read more
This paper takes a uniform look at the customized applications of proximal point algorithm (PPA) to two classes of problems: the linearly constrained convex minimization problem with a generic or separable objective function and a saddle-point problem. We model these two classes of problems uniformly by a mixed variational inequality, and show how PPA with … Read more
Recently, we have proposed to combine the alternating direction method (ADM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions without coupled variables. In this paper, we further study this topic and show … Read more
We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish … Read more
The perceptron algorithm, introduced in the late fifties in the machine learning community, is a simple greedy algorithm for finding a solution to a finite set of linear inequalities. The algorithm’s main advantages are its simplicity and noise tolerance. The algorithm’s main disadvantage is its slow convergence rate. We propose a modified version of the … Read more
In this work we consider the symmetric network equilibrium problem formulated as convex minimization problem whose variables are the path flows. In order to take into account the difficulties related to the large dimension of real network problems we adopt a column generation strategy and we employ a gradient projection method within an inexact decomposition … Read more
We consider multi-target linear-quadratic control problem on semi-infinite interval. We show that the problem can be reduced to a simple convex optimization problem on the simplex. Citation To appear in Mathematical Problems in Engineering 2012 Article Download View Multi-target Linear-quadratic control problem: semi-infinite interval
Orthogonal frequency division multiplexing (OFDM) is a technique that will prevail in the next generation wireless communication. Channel estimation is one of the key challenges in OFDM, since high-resolution channel estimation can significantly improve the equalization at the receiver and consequently enhance the communication performances. In this paper, we propose a system with an asymmetric … Read more
The old alternating direction method (ADM) has found many new applications recently, and its empirical efficiency has been well illustrated in various fields. However, the estimate of ADM’s convergence rate remains a theoretical challenge for a few decades. In this note, we provide a uniform proof to show the O(1/t) convergence rate for both the … Read more