A three-term conjugate gradient method with sufficient descent property for unconstrained optimization

Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems, because they do not need the storage of matrices. In this paper, we propose a general form of three-term conjugate gradient methods which always generate a sufficient descent direction. We give a sufficient condition for the global convergence of the proposed general method. … Read more

Extension of the semidefinite characterization of sum of squares functional systems to algebraic structures

We extend Nesterov’s semidefinite programming (SDP) characterization of the cone of functions that can be expressed as sums of squares (SOS) of functions in finite dimensional linear functional spaces. Our extension is to algebraic systems that are endowed with a binary operation which map two elements of a finite dimensional vector space to another vector … Read more

A Spectral Algorithm for Improving Graph Partitions

In the cut-improvement problem, we are asked, given a starting cut (T,V\T) in a graph G, to find a cut with low conductance around(T, V\T) or produce a certificate that there is none. More precisely, for a notion of correlation between cuts, cut-improvement algorithms seek to produce approximation guarantees of the following form: for any … Read more

Multidisciplinary Free Material Optimization

We present a mathematical framework for the so-called multidisciplinary free material optimization (MDFMO) problems, a branch of structural optimization in which the full material tensor is considered as a design variable. We extend the original problem statement by a class of generic constraints depending either on the design or on the state variables. Among the … Read more

A Limited Memory Steepest Descent Method

The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai-Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage … Read more

A Proximal Algorithm with Quasi Distance. Application to Habit’s Formation

We consider a proximal algorithm with quasi distance applied to nonconvex and nonsmooth functions involving analytic properties for an unconstrained minimization problem. We show the behavioral importance of this proximal point model for habit’s formation in Decision and Making Sciences. Article Download View A Proximal Algorithm with Quasi Distance. Application to Habit's Formation

Using approximate secant equations in limited memory methods for multilevel unconstrained optimization

The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behaviour of the objective function. Following earlier work by Gratton and Toint (2009), we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take … Read more

Enclosing Ellipsoids and Elliptic Cylinders of Semialgebraic Sets and Their Application to Error Bounds in Polynomial Optimization

This paper is concerned with a class of ellipsoidal sets (ellipsoids and elliptic cylinders) in the m-dimensional Euclidean space which are determined by a freely chosen positive semidefinite matrix. All ellipsoidal sets in this class are similar to each other through a parallel transformation and a scaling around their centers by a constant factor. Based … Read more