For a semialgebraic set K in R^n, let P_d(K) be the cone of polynomials in R^n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary. When K=R^n and d is even, we show that its boundary lies on the irreducible hypersurface defined by the discriminant of … Read more
The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to … Read more
In this paper we propose new methods for solving huge-scale optimization problems. For problems of this size, even the simplest full-dimensional vector operations are very expensive. Hence, we propose to apply an optimization technique based on random partial update of decision variables. For these methods, we prove the global estimates for the rate of convergence. … Read more
We show that the Laplace approximation of a supremum by $L^p$-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem $P$ and its dual $P^*$ appear naturally when using this simple approximation technique for the value function $g$ of $P$ or its Legendre-Fenchel conjugate $g^*$. In addition, … Read more
This paper studies stability of error bounds for convex constraint systems in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems … Read more
Total variation (TV) regularization is popular in image restoration and reconstruction due to its ability to preserve image edges. To date, most research activities on TV models concentrate on image restoration from blurry and noisy observations, while discussions on image reconstruction from random projections are relatively fewer. In this paper, we propose, analyze, and test … Read more
We consider the problem $\min_{x\in\R^n}\{f(x)\mid Ax=b, \ x\in\overline{C},\ g_j(x)\le0,\ j=1,\ldots,s\}$, where $b\in\R^m$, $A\in\R^{m\times n}$ is a full rank matrix, $\overline{C}$ is the closure of a nonempty, open and convex subset $C$ of $\R^n$, and $g_j(\cdot)$, $ j=1,\ldots,s$, are nonlinear convex functions. Our strategy consists firstly in to introduce a barrier-type penalty for the constraints $g_j(x)\le0$, … Read more
Many applications arising in a variety of fields can be well illustrated by the task of recovering the low-rank and sparse components of a given matrix. Recently, it is discovered that this NP-hard task can be well accomplished, both theoretically and numerically, via heuristically solving a convex relaxation problem where the widely-acknowledged nuclear norm and … Read more
We show that the Hedge Algorithm, a method that is widely used in Machine Learning, can be interpreted as a particular subgradient algorithm for minimizing a well-chosen convex function, namely as a Mirror Descent Scheme. Using this reformulation, we establish three modificitations and extensions of the Hedge Algorithm that are better or at least as … Read more
In this paper, we propose and study the use of alternating direction algorithms for several $\ell_1$-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis-pursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from … Read more