“Backward Differential Flow” May Not Converge to a Global Minimizer of Polynomials

We provide a simple counter-example to prove and illustrate that the backward differential flow approach, proposed by Zhu, Zhao and Liu for finding a global minimizer of coercive even-degree polynomials, can converge to a local minimizer rather than a global minimizer. We provide additional counter-examples to stress that convergence to a local minimum via the backward differential flow method is not a rare occurence. © 2015, Springer Science+Business Media New York

SSDB spaces and maximal monotonicity

In this paper, we develop some of the theory of SSD spaces and SSDB spaces, and deduce some results on maximally monotone multifunctions on a reflexive Banach space.Comment: 16 pages. Written version of the talk given at IX ISORA in Lima, Peru, October 200

A primal dual modified subgradient algorithm with sharp Lagrangian

Nonsmooth optimization, Nonconvex optimization, Duality scheme, Sharp Lagrangian, Modified subgradient algorithm, 90C26, 49M29, 49M37,

An inexact modified subgradient algorithm for nonconvex optimization

Nonconvex optimization, Nonsmooth optimization, Sharp augmented Lagrangian, Modified subgradient method, Inexact minimization, Bang-bang control,

Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach. © 2010 Springer Science+Business Media, LLC

Monotone operators without enlargements

Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized nonenlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the sum of two maximally monotone linear relations. We also present a new proof of the maximality of the sum of a maximally monotone linear relation and a normal cone operator when the domain of the linear relation intersects the interior of the domain of the normal cone

Convergence of direct methods for paramonotone variational inequalities

Monotone variational inequalities, Maximal monotone operators, Projection method,