We consider a class of infeasible, path-following methods for convex quadratric programming. Our methods are designed to be effective for solving both nondegerate and degenerate problems, where degeneracy is understood to mean the failure of strict complementarity at a solution. Global convergence and a polynomial bound on the number of iterations required is given. An … Read more
We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert space. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms, accepted for publication, DOI 10.1007/s11075-011-9490-5]. The SCNPP with only two set-valued … Read more
Many derivative-free methods for constrained problems are not efficient for minimizing functions on “thin” domains. Other algorithms, like those based on Augmented Lagrangians, deal with thin constraints using penalty-like strategies. When the constraints are computationally inexpensive but highly nonlinear, these methods spend many potentially expensive objective function evaluations motivated by the difficulties of improving feasibility. … Read more
We present and discuss a method to relax sets described by finitely many smooth convex inequality constraints by the level set of a single smooth convex inequality constraint. Based on error bounds and Lipschitz continuity, special attention is paid to the maximal approximation error and a guaranteed safety margin. Our results allow to safely avoid … Read more
In this paper we consider minimization problems with constraints. We will show that if the set of constraints is a Finslerian manifold of non positive flag curvature, and the objective function is di fferentiable and satisfi es the property Kurdyka-Lojasiewicz, then the proximal point method is naturally extended to solve that class of problems. We will prove … Read more
Simple modifiations of the limited-memory BFGS method (L-BFGS) for large scale unconstrained optimization are considered, which consist in corrections (derived from the idea of conjugate directions) of the used difference vectors, utilizing information from the preceding iteration. In case of quadratic objective functions, the improvement of convergence is the best one in some sense and … Read more
In this paper we present an iterative algorithm for the solution of regularization problems arising in inverse image processing. The regularization function to be minimized is constituted by two terms, a data fit function and a regularization function, weighted by a regularization parameter. The proposed algorithm solves the minimization problem and estimates the regularization parameter … Read more
We propose a Newton method for solving smooth unconstrained vector optimization problems under partial orders induced by general closed convex pointed cones. The method extends the one proposed by Fliege, Grana Drummond and Svaiter for multicriteria, which in turn is an extension of the classical Newton method for scalar optimization. The steplength is chosen by … Read more
The Nelder-Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalar-valued function $f$ of $n$ real variables using only function values, without any derivative information. Each Nelder–Mead iteration is associated with a nondegenerate simplex defined by $n + 1$ vertices and their function values; a typical … Read more
We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Depen- dence (RCPLD) constraint qualification. RCPLD is based on the assump- tion that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set … Read more