In this paper branching for attacking MILP is investigated. Under certain circumstances branches can be done concurrently. By introducing a new calculus it is shown there are restrictions for certain dual values and reduced costs. As a second unexpected result of this study a new class of cuts for MILP is found, which are defined … Read more
Pochet and Wolsey [Y. Pochet, L.A. Wolsey, Integer knapsack and flow covers with divisible coefficients: polyhedra, optimization and separation. Discrete Applied Mathematics 59(1995) 57-74] introduced partition inequalities for three substructures arising in various mixed integer programs, namely the integer knapsack set with nonnegative divisible/arbitrary coefficients and two forms of single-node capacitated flow set with divisible … Read more
We investigate the multi-trip elementary shortest path problem (MESPPRC) with resource constraints in which the objective is to find a shortest path between a source node and a sink node such that nodes other than the specified replenishment node are visited at most once and resource constraints are not violated. After each visit to the … Read more
We give an explicit characterization of all extreme rays of the cone of 5×5 copositive matrices. The results are based on the work of Baumert [L. D. Baumert, “Extreme copositive quadratic forms”, PhD thesis, 1965], where an implicit characterization was given. We show that the class of extreme rays found by Baumert forms a 10-dimensional … Read more
In this paper, we propose a new dwindling multidimensional filter second-order line search method for solving large-scale unconstrained optimization problems. Usually, the multidimensional filter is constructed with a fixed envelope, which is a strict condition for the gradient vectors. A dwindling multidimensional filter technique, which is a modification and improvement of the original multidimensional filter, … Read more
There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called … Read more
Accurate measurements of snow water equivalent (SWE) is an important factor in managing water resources for hydroelectric power generation. SWE over a catchment area may be estimated via kriging on measures obtained by snow monitoring devices positioned at strategic locations. The question studied in this paper is to find the device locations that minimize the … Read more
The classical trust-region subproblem (TRS) minimizes a nonconvex quadratic objective over the unit ball. In this paper, we consider extensions of TRS having extra constraints. When two parallel cuts are added to TRS, we show that the resulting nonconvex problem has an exact representation as a semidefinite program with additional linear and second-order-cone constraints. For … Read more
A new nonlinear conjugate gradient method, based on Perry’s idea, is presented. And it is shown that its sufficient descent property is independent of any line search and the eigenvalues of $P_{k+1}^{\T}P_{k+1}$ are bounded above, where $P_{k+1}$ is the iteration matrix of the new method. Thus, the global convergence is proven by the spectral analysis … Read more
We consider a derivative-free optimization, and in particular black box optimization, where the functions to be minimized and the functions representing the constraints are given by black boxes without derivatives. Two fundamental families of methods are available: model-based methods and directional direct search algorithms. This work exploits the flexibility of the second type of methods … Read more