In this paper, we propose and analyze a trust-region model-based algorithm for solving unconstrained stochastic optimization problems. Our framework utilizes random models of an objective function $f(x)$, obtained from stochastic observations of the function or its gradient. Our method also utilizes estimates of function values to gauge progress that is being made. The convergence analysis … Read more
We show that for any positive integer $d$, there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree $\leq d$, or (ii) a polytopic Lyapunov function with $\leq d$ facets, or (iii) a piecewise … Read more
We develop subgradient- and gradient-based methods for minimizing strongly convex functions under a notion which generalizes the standard Euclidean strong convexity. We propose a unifying framework for subgradient methods which yields two kinds of methods, namely, the Proximal Gradient Method (PGM) and the Conditional Gradient Method (CGM), unifying several existing methods. The unifying framework provides … Read more
In this paper we address the stable numerical solution of nonlinear ill-posed systems by a trust-region method. We show that an appropriate choice of the trust-region radius gives rise to a procedure that has the potential to approach a solution of the unperturbed system. This regularizing property is shown theoretically and validated numerically. CitationDipartimento di … Read more
We investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to … Read more
For the minimization of a nonlinear cost functional under convex constraints the relaxed projected gradient process is a well known method. The analysis is classically performed in a Hilbert space. We generalize this method to functionals which are differentiable in a Banach space. The search direction is calculated by a quadratic approximation of the cost … Read more
Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate … Read more
In this paper, we further establish two types of DC (Difference of Convex functions) programming for $l_0$ sparse reconstruction. Our DC objective functions are specified to the difference of two norms. One is the difference of $l_1$ and $l_{\sigma_q}$ norms (DC $l_1$-$l_{\sigma_q}$ for short) where $l_{\sigma_q}$ is the $l_1$ norm of the $q$-term ($q\geq1$) best … Read more
This paper presents a reformulation that is a natural “by-product” of the ‘variable endogenization’ process for equality-constrained programs. The method results a partial relaxation of the constraints which in turn confers some computational advantages. A fully-annotated example illustrates the technique and presents some comparative numerical results. CitationSiwale, I.: Partial Relaxation of Equality-constrained Programs. Technical Report … Read more
Given the bilinear function f(x,y) = xy, and the constraint x