Convex quadratic programming with one constraint and bounded variables

doi:10.1007/BF02591992

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Convex quadratic programming with one constraint and bounded variables

Journal Article•DOI•

Convex quadratic programming with one constraint and bounded variables

Jean-Pierre Dussault¹, Jacques A. Ferland², Bernard Lemaire³•Institutions (3)

Université de Sherbrooke¹, Université de Montréal², University of Montpellier³

01 Oct 1986-Mathematical Programming (Springer-Verlag)-Vol. 36, Iss: 1, pp 90-104

TL;DR: An iterative algorithm for solving a convex quadratic program with one equality constraint and bounded variables and preliminary testing suggests that this approach is efficient for problems with diagonally dominant matrices.

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Abstract: In this paper we propose an iterative algorithm for solving a convex quadratic program with one equality constraint and bounded variables. At each iteration, a separable convex quadratic program with the same constraint set is solved. Two variants are analyzed: one that uses an exact line search, and the other a unit step size. Preliminary testing suggests that this approach is efficient for problems with diagonally dominant matrices.

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Journal Article•DOI•

Demand Estimation and Assortment Optimization Under Substitution: Methodology and Application

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A. Gürhan Kök¹, Marshall L. Fisher²•Institutions (2)

Duke University¹, University of Pennsylvania²

01 Nov 2007-Operations Research

TL;DR: An algorithmic process to help retailers compute the best assortment for each store and establishes new structural properties that relate the products included in the assortment and their inventory levels to product characteristics such as gross margin, case-pack sizes, and demand variability.

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Abstract: Assortment planning at a retailer entails both selecting the set of products to be carried and setting inventory levels for each product. We study an assortment planning model in which consumers might accept substitutes when their favorite product is unavailable. We develop an algorithmic process to help retailers compute the best assortment for each store. First, we present a procedure for estimating the parameters of substitution behavior and demand for products in each store, including the products that have not been previously carried in that store. Second, we propose an iterative optimization heuristic for solving the assortment planning problem. In a computational study, we find that its solutions, on average, are within 0.5% of the optimal solution. Third, we establish new structural properties (based on the heuristic solution) that relate the products included in the assortment and their inventory levels to product characteristics such as gross margin, case-pack sizes, and demand variability. We applied our method at Albert Heijn, a supermarket chain in The Netherlands. Comparing the recommendations of our system with the existing assortments suggests a more than 50% increase in profits.

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419 citations

Cites background or methods from "Convex quadratic programming with o..."

...Dussault et al. (1986) and Klastorin (1990) approximate the quadratic problem with a series of separable problems....
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...For nonseparable resource allocation problems, another iterative method guaranteed to reach the optimal solution in quadratic optimization problems is found in Dussault et al. (1986) and Klastorin (1990). They simply solve the integer problem with the branch-and-bound method, but at each node of the branch-and-bound, method, solve the relaxed (i....
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...For nonseparable resource allocation problems, another iterative method guaranteed to reach the optimal solution in quadratic optimization problems is found in Dussault et al. (1986) and Klastorin (1990)....
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...For nonseparable resource allocation problems, another iterative method guaranteed to reach the optimal solution in quadratic optimization problems is found in Dussault et al. (1986) and Klastorin (1990). They simply solve the integer problem with the branch-and-bound method, but at each node of the branch-and-bound, method, solve the relaxed (i.e., continuous variables) nonseparable quadratic optimization problem with a series of separable problems. At each iteration, Taylor approximation is updated based on the current solution. Our method applies a similar idea to the discrete space. The application of the greedy heuristic in Step 3 corresponds to the solution of the approximate separable objective function, and the updating of the demand vector using the current solution corresponds to the recomputation of the Taylor approximation. The difference is that our method incorporates integrality at each iteration, whereas Dussault et al. (1986) and Klastorin (1990) impose integrality constraints at the highest level of hierarchy in the branch-and-bound method....
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...The difference is that our method incorporates integrality at each iteration, whereas Dussault et al. (1986) and Klastorin (1990) impose integrality constraints at the highest level of hierarchy in the branch-and-bound method....
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Journal Article•DOI•

The nonlinear knapsack problem – algorithms and applications

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Kurt M. Bretthauer¹, Bala Shetty²•Institutions (2)

Indiana University¹, Texas A&M University²

01 May 2002-European Journal of Operational Research

TL;DR: A survey of algorithms and applications for the nonlinear knapsack problem, a nonlinear optimization problem with just one constraint, bounds on the variables, and a set of specially structured constraints such as generalized upper bounds (GUBs), is presented.

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230 citations

Journal Article•DOI•

A survey on the continuous nonlinear resource allocation problem

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Michael Patriksson¹•Institutions (1)

Chalmers University of Technology¹

16 Feb 2008-European Journal of Operational Research

TL;DR: This paper surveys the history and applications of the problem, as well as algorithmic approaches to its solution, and analyzes the most relevant references, especially regarding their originality and numerical findings.

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204 citations

Proceedings Article•DOI•

Multi-resource allocation: Fairness-efficiency tradeoffs in a unifying framework

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Carlee Joe-Wong¹, Soumya Sen¹, Tian Lan², Mung Chiang¹•Institutions (2)

Princeton University¹, George Washington University²

25 Mar 2012

TL;DR: Two families of fairness functions are developed that provide different tradeoffs, characterize the effect of user requests' heterogeneity, and prove conditions under which these fairness measures satisfy the Pareto efficiency, sharing incentive, and envy-free properties.

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Abstract: Quantifying the notion of fairness is under-explored when users request different ratios of multiple distinct resource types. A typical example is datacenters processing jobs with heterogeneous resource requirements on CPU, memory, etc. A generalization of max-min fairness to multiple resources was recently proposed in [1], but may suffer from significant loss of efficiency. This paper develops a unifying framework addressing this fairness-efficiency tradeoff with multiple resource types. We develop two families of fairness functions which provide different tradeoffs, characterize the effect of user requests' heterogeneity, and prove conditions under which these fairness measures satisfy the Pareto efficiency, sharing incentive, and envy-free properties. Intuitions behind the analysis are explained in two visualizations of multi-resource allocation.

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203 citations

Journal Article•DOI•

Competitive facility location model with concave demand

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Robert Aboolian¹, Oded Berman², Dmitry Krass²•Institutions (2)

California State University San Marcos¹, University of Toronto²

01 Sep 2007-European Journal of Operational Research

TL;DR: A spatial interaction model for locating a set of new facilities that compete for customer demand with each other, as well as with some pre-existing facilities to capture the “market expansion” and the "market cannibalization" effects is considered.

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106 citations

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Journal Article•DOI•

Constrained minimization methods

[...]

E.S. Levitin, B.T. Polyak

01 Jan 1966-Ussr Computational Mathematics and Mathematical Physics

785 citations

Journal Article•DOI•

Convex programming in Hilbert space

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A. A. Goldstein

01 Sep 1964-Bulletin of the American Mathematical Society

TL;DR: In this article, the authors give a construction for minimizing certain twice-differentiable functions on a closed convex subset C, of a Hubert Space, H. The algorithm assumes one can constructively project points onto convex sets.

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Abstract: This note gives a construction for minimizing certain twice-differentiable functions on a closed convex subset C, of a Hubert Space, H. The algorithm assumes one can constructively \"project\" points onto convex sets. A related algorithm may be found in Cheney-Goldstein [ l ] , where a constructive fixed-point theorem is employed to construct points inducing a minimum distance between two convex sets. In certain instances when such projections are not too difficult to construct, say on spheres, linear varieties, and orthants, the method can be effective. For applications to control theory, for example, see Balakrishnan [2], and Goldstein [3]. In what follows P will denote the \"projection\" operator for the convex set C. This operator, which is well defined and Lipschitzian, assigns to a given point in H its closest point in C (see, e.g., [ l ] ) . Take x £ f f and y £ C . Then [x — y, P(x) —y]^\\\\P(x) —y\\\\. In the nontrivial case this inequality is a consequence of the fact that C is supported by a hyperplane through P(x) with normal x — P(x). Let ƒ be a real-valued function on H and x0 an arbitrary point of C. Let 5 denote the level set (xGC: / (x ) ^f(x0)}, and let S be any open set containing the convex hull of S. Let ƒ'(*, • )= [V/(x), •] signify the Fréchet derivative of ƒ at x. A point zin C will be called stationary if P(z—pVf(z)) =z for all p > 0 ; equivalently, when ƒ is convex the linear functional f (z, •) achieves a minimum on C a t z.

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564 citations

Journal Article•DOI•

On the Goldstein-Levitin-Polyak gradient projection method

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Dimitri P. Bertsekas¹•Institutions (1)

University of Illinois at Urbana–Champaign¹

01 Nov 1974

TL;DR: In this paper, the authors consider the convergence of a gradient projection method to a conjugate direction with a quasi-Newton or Newton's method and achieve the attendant superlinear convergence rate.

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Abstract: This paper considers some aspects of a gradient projection method proposed by Goldstein [1], Levitin and Polyak [3], and more recently, in a less general context, by McCormick [10]. We propose and analyze some convergent step-size rules to be used in conjunction with the method. These rules are similar in spirit to the efficient Armijo rule for the method of steepest descent and under mild assumptions they have the desirable property that they identify the set of active inequality constraints in a finite number of iterations. As a result the method may be converted towards the end of the process to a conjugate direction, quasi-Newton or Newton's method, and achieve the attendant superlinear convergence rate. As an example we propose some quadratically convergent combinations of the method with Newton's method. Such combined methods appear to be very efficient for large-scale problems with many simple constraints such as those often appearing in optimal control.

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524 citations

Journal Article•DOI•

Introduction a l'analyse numerique matricielle et a l'optimisation

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Philippe G. Ciarlet¹•Institutions (1)

Pierre-and-Marie-Curie University¹

01 Apr 1984-Mathematics of Computation

365 citations

Journal Article•DOI•

Finding the nearest point in A polytope

[...]

Philip Wolfe¹•Institutions (1)

IBM¹

01 Dec 1976-Mathematical Programming

TL;DR: A terminating algorithm is developed for the problem of finding the point of smallest Euclidean norm in the convex hull of a given finite point set in Euclideann-space, or equivalently for finding an “optimal” hyperplane separating a given point from a given infinite point set.

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Abstract: A terminating algorithm is developed for the problem of finding the point of smallest Euclidean norm in the convex hull of a given finite point set in Euclideann-space, or equivalently for finding an “optimal” hyperplane separating a given point from a given finite point set. Its efficiency and accuracy are investigated, and its extension to the separation of two sets and other convex programming problems described.

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331 citations