We review the superiorization methodology, which can be thought of, in some cases, as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to an objective function value) to one returned … Read more
In view of solving nonsmooth and nonconvex problems involving complex constraints (like standard NLP problems), we study general maximization-minimization procedures produced by families of strongly convex sub-problems. Using techniques from semi-algebraic geometry and variational analysis –in particular Lojasiewicz inequality– we establish the convergence of sequences generated by this type of schemes to critical points. The … Read more
The Minimum Spanning Tree Problem (MSTP) is one of the most known combinatorial optimization problems. It concerns the determination of a minimum edge-cost subgraph spanning all the vertices of a given connected graph. The Quadratic Minimum Spanning Tree Problem (QMSTP) is a variant of the MST whose cost considers also the interaction between every pair … Read more
We present a filter line-search algorithm that does not require inertia information about the linear system to ensure global convergence. The proposed approach performs curvature tests along the search step to ensure descent. This feature permits more modularity in the linear algebra, enabling the use of a wider range of iterative and decomposition strategies. We … Read more
We introduce a new method for a task of maximal material utilization, which is is to fit a flexible, scalable three-dimensional body into another aiming for maximal volume whereas position and shape may vary. The difficulty arises from the containment constraint which is not easy to handle numerically. We use a collision detection method to … Read more
We consider the problem of optimizing an unknown function given as an oracle over a mixed-integer box-constrained set. We assume that the oracle is expensive to evaluate, so that estimating partial derivatives by finite differences is impractical. In the literature, this is typically called a black-box optimization problem with costly evaluation. This paper describes the … Read more
\begin{abstract} The existing choices for the step lengths used by the classical steepest descent algorithm for minimizing a convex quadratic function require in the worst case $ \Or(C\log(1/\eps)) $ iterations to achieve a precision $ \eps $, where $ C $ is the Hessian condition number. We show how to construct a sequence of step … Read more
This paper considers computation of Fr\’echet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to … Read more
Quadratic models Q_k(.) of the objective function F(.) are used by many successful iterative algorithms for minimization, where k is the iteration number. Given the vector of variables x_k, a new vector x_{k+1} may be calculated that satisfies Q_k(x_{k+1}) < Q_k(x_k), in the hope that it provides the reduction F(x_{k+1}) < F(x_k). Trust region methods ... Read more
A new class of matrix support functionals is presented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo matrix-fractional function, and the nuclear norm. The support function is based on the graph of the product of a matrix with its transpose. Closed form expressions for the support … Read more