n-step Mingling Inequalities: New Facets for the Mixed-Integer Knapsack Set

The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi are valid inequalities for the mixed-integer knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamturk and Gunluk are also derived based on MIR and they incorporate upper bounds … Read more

Alternating directions based contraction method for generally separable linearly constrained convex programming problems

The classical alternating direction method (ADM) has been well studied in the context of linearly constrained convex programming problems and variational inequalities where both the involved operators and constraints are separable into two parts. In particular, recentness has witnessed a number of novel applications arising in diversified areas (e.g. Image Processing and Statistics), for which … Read more

Copositive Programming – a Survey

Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field … Read more

Alternating direction algorithms for total variation deconvolution in image reconstruction

Image restoration and reconstruction from blurry and noisy observation is known to be ill-posed. To stabilize the recovery, total variation (TV) regularization was introduced by Rudin, Osher and Fatemi in \cite{LIR92}, which has demonstrated superiority in preserving image edges. However, the nondifferentiability of TV makes the underlying optimization problems difficult to solve. In this paper, … Read more

On convex envelopes and underestimators for bivariate functions

In this paper we discuss convex underestimators for bivariate functions. We first present a method for deriving convex envelopes over the simplest two-dimensional polytopes, i.e., triangles. Next, we propose a technique to compute the value at some point of the convex envelope over a general two-dimensional polytope, together with a supporting hyperplane of the convex … Read more

Data Fitting and Experimental Design in Dynamical Systems with EASY-FIT ModelDesign

EASY-FIT is an interactive software system to identify parameters and compute optimal designs in explicit model functions, steady-state systems, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time dependent partial differential equations with or without algebraic equations. Proceeding from given experimental data, i.e. observation times and measurements, the minimum … Read more

An Updated Set of 306 Test Problems for Nonlinear Programming with Validated Optimal Solutions

The availability of nonlinear programming test problems is extremely important to test optimization codes or to develop new algorithms. We describe the usage of the Fortran subroutines for all 306 test problems of two previous collections of the author, see Hock and Schittkowski (1981) and Schittkowski (1987). For each test example, we provide an optimal … Read more

A new sequential optimality condition for constrained optimization and algorithmic consequences

Necessary first-order sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. These conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qual- i cations. A new strong sequential optimality condition is introduced in the present paper. A proof that a well established Augmented Lagrangian … Read more

An L1 Elastic Interior-Point Method for Mathematical Programs with Complementarity Constraints

We propose an interior-point algorithm based on an elastic formulation of the L1-penalty merit function for mathematical programs with complementarity constraints. The method generalizes that of Gould, Orban and Toint (2003) and naturally converges to a strongly stationary point or delivers a certificate of degeneracy without recourse to second-order intermediate solutions. Remarkably, the method allows … Read more

Local superlinear convergence of polynomial-time interior-point methods for hyperbolic cone optimization problems

In this paper, we establish the local superlinear convergence property of some polynomial-time interior-point methods for an important family of conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier function, which must have {\em negative curvature}. We propose a new path-following predictor-corrector scheme, which work only … Read more