Browsing by Subject "Convex hull"

Now showing 1 - 4 of 4

Open Access
Continuous knapsack sets with divisible capacities
(Springer, 2016) Wolsey, L. A.; Yaman, H.
We study two continuous knapsack sets (Formula presented.) and (Formula presented.) with (Formula presented.) integer, one unbounded continuous and (Formula presented.) bounded continuous variables in either (Formula presented.) or (Formula presented.) form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and (Formula presented.) polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable. The latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the (Formula presented.) case. We also provide an extended formulation for the (Formula presented.) case. It follows that, given a specific objective function, optimization over both (Formula presented.) and (Formula presented.) can be carried out by solving (Formula presented.) polynomial size linear programs. A further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality of the convex hull of such sets.
Open Access
Convex hull results for the warehouse problem
(Elsevier B.V., 2018) Wolsey, L. A.; Yaman, Hande
Given an initial stock and a capacitated warehouse, the warehouse problem aims to decide when to sell and purchase to maximize profit. This problem is common in revenue management and energy storage. We extend this problem by incorporating fixed costs and provide convex hull descriptions as well as tight compact extended formulations for several variants. For this purpose, we first derive unit flow formulations based on characterizations of extreme points and then project out the additional variables using Fourier-Motzkin elimination. It turns out that the nontrivial inequalities are flow cover inequalities for some single node flow set relaxations.
Open Access
Relaxation of multidimensional variational problems with constraints of general form
(Pergamon Press, 2001) Hüsseinov, F.
A relaxation of multidimensional variational problems with constraint of rather general form on gradients of admissible functions is investigated. It is assumed that the gradient of an admissible function belongs to an arbitrary bounded set. This relaxation involves as a class of admissible function the closure of the class of admissible functions of the original problem in the topology of uniform convergence, and uses a theorem characterizing this closure.
Open Access
The splittable flow arc set with capacity and minimum load constraints
(Elsevier, 2013) Yaman, H.
We study the convex hull of the splittable flow arc set with capacity and minimum load constraints. This set arises as a relaxation of problems where clients have demand for a resource that can be installed in integer amounts and that has capacity limitations and lower bounds on utilization. We prove that the convex hull of this set is the intersection of the convex hull of the set with a capacity constraint and the convex hull of the set with a minimum load constraint.