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QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints
"... . We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) co ..."
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Cited by 24 (8 self)
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. We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, illconditioning, convexity of the objective, monotonicity and symmetry of the secondlevel problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.maths.mu.OZ.AU/~danny/qpecgendoc.h...
A New Technique for Generating Quadratic Programming Test Problems
 Mathematical Programming
, 1993
"... This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum ..."
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Cited by 6 (0 self)
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This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated. Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems.
Geometry And Local Optimality Conditions For Bilevel Programs With Quadratic Strictly Convex Lower Levels
 In: D. Du, & M. Pardalos (Eds.), Minimax
, 1995
"... This paper describes necessary and sufficient optimality conditions for bilevel programming problems with quadratic strictly convex lower levels. By examining the local geometry of these problems we establish that the set of feasible directions at a given point is composed of a finite union of conve ..."
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This paper describes necessary and sufficient optimality conditions for bilevel programming problems with quadratic strictly convex lower levels. By examining the local geometry of these problems we establish that the set of feasible directions at a given point is composed of a finite union of convex cones. Based on this result, we show that the optimality conditions are simple generalizations of the first and second order optimality conditions for mathematical (one level) programming problems. 1 INTRODUCTION A bilevel program is defined as the problem of minimizing a function f (the upper level function) in two different vectors of variables x and y subject to (upper level) constraints, where the vector y is an optimal solution of another constrained optimization problem (the lower level problem) parameterized by the vector x. References [2] and [17] survey the extensive research that has been done in bilevel programming. 2 Chapter 1 It is interesting to note that any minimax probl...
Parallel Implementation of Successive Convex Relaxation Methods for . . .
 J. OF GLOBAL OPTIMIZATION
, 2002
"... As computing resources continue to improve, global solutions for larger size quadraticallyconstrained optimization problems become more achievable. In this paper, we focus on larger size problems and get accurate bounds for optimal values of such problems with the successive use of SDP relaxations ..."
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As computing resources continue to improve, global solutions for larger size quadraticallyconstrained optimization problems become more achievable. In this paper, we focus on larger size problems and get accurate bounds for optimal values of such problems with the successive use of SDP relaxations on a parallel computing system called Ninf (Network based Information Library for high performance computing).
Interdiction Branching
, 2011
"... This paper introduces interdiction branching, a new branching method for binary integer programs that is designed to overcome the difficulties encountered in solving problems for which branching on variables is inherently weak. Unlike traditional methods, selection of the disjunction in interdiction ..."
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This paper introduces interdiction branching, a new branching method for binary integer programs that is designed to overcome the difficulties encountered in solving problems for which branching on variables is inherently weak. Unlike traditional methods, selection of the disjunction in interdiction branching takes into account the best feasible solution found so far. In particular, the method is based on computing an improving solution cover, which is a set of variables of which at least one must be nonzero in any improving solution. From an improving solution cover, we can obtain a branching disjunction with desirable properties. Any minimal such cover yields a disjunction in which multiple variables are fixed in each child node and for which each child node is guaranteed to contain at least one improving solution. Computing a minimal improving solution cover amounts to solving a discrete bilevel program, which is difficult in general. In practice, a solution cover, although not necessarily minimal nor improving, can be found using a heuristic that achieves a profitable tradeoff between the size of the enumeration tree and the computational burden of computing the cover. An empirical study on a test suite of difficult binary knapsack and stable set problems shows that an implementation of the method dramatically reduces the size of the enumeration tree compared to branching on variables, yielding significant savings in running times.
Successive Convex Relaxation Approach to Bilevel Quadratic Optimization Problems
 Dept
, 1999
"... . The quadratic bilevel programming problem is an instance of a quadratic hierarchical decision process where the lower level constraint set is dependent on decisions taken at the upper level. By replacing the inner problem by its corresponding KKT optimality conditions, the problem is transformed t ..."
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Cited by 1 (1 self)
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. The quadratic bilevel programming problem is an instance of a quadratic hierarchical decision process where the lower level constraint set is dependent on decisions taken at the upper level. By replacing the inner problem by its corresponding KKT optimality conditions, the problem is transformed to a single yet nonconvex quadratic program, due to the complementarity condition. In this paper we adopt the successive convex relaxation approach proposed by Kojima and Tun¸cel for computing a convex relaxation of a nonconvex feasible region. By further exploiting the special structure of the bilevel problem, we establish new techniques which enable the efficient implementation of the proposed algorithm. The performance of these techniques is tested in a comparison with other procedures using a number of test problems of quadratic bilevel programming. 1 Introduction. Bilevel programming (abbreviated by BP) belongs to a class of nonconvex global optimization problems. It arises where decis...
OPTIMIZATION
, 2004
"... A multilevel parallelized branch and bound algorithm for quadratic optimization ..."
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A multilevel parallelized branch and bound algorithm for quadratic optimization
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"... QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints∗ ..."
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QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints∗
References
"... We wish to study the following type of twoplayer game: Scenario There are two players, a leader and a follower. Each player controls the level of certain activities subject to constraints and seeks to optimize a given objective function. The leader sets activity levels first. The follower then reac ..."
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We wish to study the following type of twoplayer game: Scenario There are two players, a leader and a follower. Each player controls the level of certain activities subject to constraints and seeks to optimize a given objective function. The leader sets activity levels first. The follower then reacts based on the choices of the leader. The leader can affect the follower through constraints on the follower that involve the leader’s choices. This is an example of what are generally known in the literature as Stackelberg games. Here, we consider static games in which there is only one round of decisionmaking. We can model such games mathematically as bilevel programs.