Allocation of limited resources under quadratic constraints

arXiv (Cornell University), 2018

We use an algebraic viewpoint, namely a matrix framework to deal with the problem of resource allocation under uncertainty in the context of a qualitative approach. Our basic qualitative data are a plausibility relation over the resources, a hierarchical relation over the agents and of course the preference that the agents have over the resources. With this data we propose a qualitative binary relation between allocations such that F G has the following intended meaning: the allocation F produces more or equal social welfare than the allocation G. We prove that there is a family of allocations which are maximal with respect to. We prove also that there is a notion of simple deal such that optimal allocations can be reached by sequences of simple deals. Finally, we introduce some mechanism for discriminating optimal allocations.

Mathematical Methods of Operations Research, 2001

Mathematical programming problems with equilibrium constraints (MPEC) are nonlinear programming problems where the constraints have a form that is analogous to first-order optimality conditions of constrained optimization. We prove that, under reasonable sufficient conditions, stationary points of the sum of squares of the constraints are feasible points of the MPEC. In usual formulations of MPEC all the feasible points are nonregular in the sense that they do not satisfy the Mangasarian-Fromovitz constraint qualification of nonlinear programming. Therefore, all the feasible points satisfy the classical Fritz-John necessary optimality conditions. In principle, this can cause serious difficulties for nonlinear programming algorithms applied to MPEC. However, we show that most feasible points do not satisfy a recently introduced stronger optimality condition for nonlinear programming. This is the reason why, in general, nonlinear programming algorithms are successful when applied to MPEC.

2017

Le contenu de cette these est divise en deux parties. La premiere partie de cette these porte sur l'etude d'approches heuristiques pour approximer des fronts de Pareto. Nous proposons un nouvel algorithme de recherche locale pour resoudre des problemes d'optimisation combinatoire. Cette technique est integree dans un modele operationnel generique ou l'algorithme evolue vers de nouvelles solutions formees en combinant des solutions trouvees dans les etapes precedentes. Cette methode ameliore les algorithmes de recherche locale existants pour resoudre le probleme d'assignation quadratique bi- et tri-objectifs.La seconde partie se focalise sur les algorithmes d'ordonnancement dans un contexte non-preemptif. Plus precisement, nous etudions le probleme de la minimisation du stretch maximum sur une seule machine pour une execution online. Nous presentons des resultats positifs et negatifs, puis nous donnons une solution optimale semi-online. Nous etudions ensuite l...

Discrete Applied Mathematics, 1990

it is necessary to find a minimum weight assignment that satisfies one or several additional resource constraints. For example, consider the problem of assigning persons to jobs where each assignment utilizes at least two scarce resources and the resource utilization is dependent on the person and the type of task. A practical situation where the above might occur is a slaughter house where the "cutters" are assigned to different cut patterns. In this case the resources are the time, the cost and the productivity measured in terms of quality and amount of the end products.

SSRN Electronic Journal, 2000

We revisit the problem of minimizing a separable convex function with a linear constraint and box constraints. This optimization problem arises naturally in many applications in economics, finance and insurance. Existing literature exclusively tackles this problem by using the traditional Kuhn-Tucker theory, which leads to either iterative schemes or yields explicit solutions only under some special classes of convex functions. This paper presents a new approach of solving this constrained minimization problem by using the theory of comonotonicity. The key step is to apply an integral representation result to express each convex function as the expected stop-loss of some suitable random variable. By using this approach, we can derive not only the explicit solution, but also obtain its geometric meaning and some other qualitative properties such as uniqueness.

2006

In order to solve more easily combinatorial optimization problems, one way is to find theoretically a set of necessary or/and sufficient conditions that the optimal solution of the problem has to verify. For instance, in linear programming, weak and strong duality conditions can be easily derived. And in convex quadratic programming, Karush-Kuhn-Tucker conditions gives necessary and sufficient conditions. Despite that, in general, such conditions doesn't exist for integer programming, some necessary conditions can be derived from Karush-Kuhn-Tucker conditions for the unconstrained quadratic 0-1 problem. In this paper, we present these conditions. We show how they can be used with constraints programming techniques to fix directly some variables of the problem. Hence, reducing strongly the size of the problem. For example, for low density problems of size lower than 50, those conditions combined with constraints programming may be sufficient to solve completely the problem, without branching. In general, for quadratic 0-1 problems with linear constraints, we propose a new method combining these conditions with constraints and linear programming. Some numerical results, with the instances of the OR-Library, will be presented.

In this paper 1 , we present a constraint programmingbased approach to solve a hard real-time allocation problem. This problem consists in assigning periodic tasks to processors in the context of fixed priority preemptive scheduling. Our approach builds on dynamic constraint programming together with a learning method to find a feasible processor allocation under constraints. This problem is decomposed into two subproblems: allocation, and schedulability. Benders decomposition is then used as a way of learning when the allocation subproblem yields a valid solution while the schedulability analysis of the allocation does not. The rationale of this approach is to learn from the failures of the schedulability analysis to reduce the search space.

2006

In order to solve more easily combinatorial optimization problems, one way is to find theoretically a set of necessary or/and sufficient conditions that the optimal solution of the problem has to verify. For instance, in linear programming, weak and strong duality conditions can be easily de- rived. And in convex quadratic programming, Karush-Kuhn-Tucker conditions gives necessary and sufficient conditions. Despite that, in general, such conditions doesn’t exist for integer programming, some necessary conditions can be derived from Karush-Kuhn-Tucker conditions for the unconstrained quadratic 0-1 problem. In this paper, we present these conditions. We show how they can be used with constraint program- ming techniques to fix directly some variables of the problem. Hence, reducing strongly the size of the problem. For example, for low density problems of size lower than 50, those conditions combined with constraint programming may be sufficient to solve completely the problem, without...

Decision Sciences, 1974

This paper is concerned with the solution of linear and linear goal programming problems in which the values of the right-hand side parameters are not fixed constants. Specifically, we are concerned with linear optimization problems in which the right-hand sides of the constraining equations are free to vary subject to a set of linear constraining equations. By formulating a relaxed linear program wherein the right-hand sides are treated as variables, we show how it is possible to solve one larger linear program that yields as a solution not only the optimal values for the decision variables, but also the optimal values for the right-hand sides.

2000

Optimization Models for Resource allocation are investing in how to make the best use of available but limited resources in order to achieve the best results. In strategic planning, resource allocation is a plan for using available resources, especially in the near future, to achieve the goals of the future. It is a process of allocating resources during the entire planning horizon and among the various units. Resource allocation plans can be decided by using mathematical programming. In this dissertation, the research has been focused on how to allocate resources in the uncertain environment. The mathematical programming formulations for the resource allocation model under severe uncertainty will be studied. In particular, we will focus on solving the stability issues of the traditional probabilistic model. We propose an approach consisting of solving a sequence of convex robust optimization models with unknown-but-bounded random variables along with the stochastic programming to pursue the allocation performance for the expected overall objective value. Our theoretical results show that the proposed approach can always obtain an equivalent or a better expected revenue with the corresponding allocation, while significantly reducing the risk under perturbations. Although this method requires solving two convex mathematical v programming models, both models are solved within a timely manner thanks to their convex model instances and with effective, and less, computationally demanding algorithms. With the increasing threats from public health emergencies, such as earthquakes, tornados, pandemic flus, or terrorist attacks, high attention has been paid to the public health response to a pandemic from federal to national level, together with local health departments, and the health-care community. Various organizations cooperate with each other to strengthen the preparedness for the pandemic and disastrous emergencies, thus to improve the public health. The Strategic National Stockpile (SNS) is maintained by the Centers for Disease Control and Prevention (CDC) and the U.S. Department of Health and Human Services (DHHS) for the United States in the event of a shortage of local medical resources or other unanticipated supply problems. The SNS is the United States' national repository of antibiotics, vaccines, chemical antidotes, antitoxins, and other critical medical equipment and supplies. It has the capability to supplement or re-supply local health authorities with necessary materials for relief action within the response time in as little as 12 hours. The pilot study is done with the support of Kentucky SNS to determine the capacity allocation plan for each county in order to maximize the health benefit under various uncertainties, which can never be accurately estimated. We thereby employ a heuristic method named "resource reservation" to suggest the resource allocation plan for Kentucky SNS. vi