Examples exist of extended-real-valued closed functions on $\R^n$ whose subdifferentials (in the standard, limiting sense) have large graphs. By contrast, if such a function is semi-algebraic, then its subdifferential graph must have everywhere constant local dimension $n$. This result is related to a celebrated theorem of Minty, and surprisingly may fail for the Clarke subdifferential. … Read more
We discuss two nonsmooth functions on R^n introduced by Nesterov. We show that the first variant is partly smooth in the sense of [A.S. Lewis. Active sets, nonsmoothness and sensitivity. SIAM Journal on Optimization, 13:702–725, 2003.] and that its only stationary point is the global minimizer. In contrast, we show that the second variant has … Read more
The $\epsilon$-pseudospectral abscissa $\alpha_\epsilon$ and radius $\rho_\epsilon$ of an n x n matrix are respectively the maximal real part and the maximal modulus of points in its $\epsilon$-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang. Variational analysis of pseudospectra. SIAM Journal on Optimization, 19:1048-1072, 2008] that for fixed … Read more
The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers … Read more
This paper introduces a novel algorithm for the nonnegative matrix factorization and completion problem, which aims to nd nonnegative matrices X and Y from a subset of entries of a nonnegative matrix M so that XY approximates M. This problem is closely related to the two existing problems: nonnegative matrix factorization and low-rank matrix completion, … Read more
It’s shown that a number of variational problems can be cast as finding the maxinf-points (or minsup-points) of bivariate functions, coveniently abbreviated to bifunctions. These variational problems include: linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, non-cooperative games, Walras and Nash equilibrium problems. One can then appeal to the theory of lopsided convergence … Read more
The paper describes a new primal relaxation (PR) for computing bounds on nonlinear integer programming (NLIP) problems. It is a natural extension to NLIP problems of the geometric interpretation of Lagrangean relaxation presented by Geoffrion (1974) for linear problems, and it is based on the same assumption that some constraints are complicating and are treated … Read more
The Kantorovich function $(x^TAx)( x^T A^{-1} x)$, where $A$ is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we prove that the 2-dimensional Kantorovich function is convex if and only … Read more
In this paper, we establish new bounds for restricted isometry constants (RIC) in low-rank matrix recovery. Let $\A$ be a linear transformation from $\R^{m \times n}$ into $\R^p$, and $r$ the rank of recovered matrix $X\in \R^{m \times n}$. Our main result is that if the condition on RIC satisfies $\delta_{2r+k}+2(\frac{r}{k})^{1/2}\delta_{\max\{r+\frac{3}{2}k,2k\}}
In this paper, we propose an efficient approach for solving a class of convex optimization problems in Hilbert spaces. Our feasible region is a (possibly infinite-dimensional) simple convex set, i.e. we assume that projections on this set are computationally easy to compute. The problem we consider is the minimization of a convex function over this … Read more