A PRINCIPAL PIVOTING ALGORITHM FOR LINEAR AND QUADRATIC PROGRAMS

2015 ◽  
pp. 597-597
Keyword(s):  
Quadratic Programs ◽  
Pivoting Algorithm ◽  
Principal Pivoting

A block principal pivoting algorithm for vertical generalized LCP with a vertical block P-matrix

2021 ◽  
pp. 113913
Author(s):  
Aniekan A. Ebiefung ◽  
Luís M. Fernandes ◽  
Joaquim J. Júdice ◽  
Michael M. Kostreva
Keyword(s):  
Pivoting Algorithm ◽  
Principal Pivoting ◽  
P Matrix

On Solving Mean Payoff Games Using Pivoting Algorithms

2018 ◽  
Vol 35 (05) ◽  
pp. 1850035
Author(s):  
S. K. Neogy ◽  
Prasenjit Mondal ◽  
Abhijit Gupta ◽  
Debasish Ghorui
Keyword(s):  
Stochastic Games ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Lemke's Algorithm ◽  
Counter Example ◽  
Pivoting Algorithm ◽  
The Matrix ◽  
Principal Pivoting ◽  
Mean Payoff Games ◽  
Mean Payoff

Two classical pivoting algorithms, due to Lemke and Cottle–Dantzig, are studied on linear complementarity problems (LCPs) and their generalizations that arise from infinite duration two-person mean payoff games (MPGs) under zero-mean partition problem. Lemke’s algorithm was studied in solving MPGs via reduction to discounted payoff games or to simple stochastic games. We provide an alternative and efficient approach for studying the LCPs arising from the MPGs without any reduction. A binary MPG can easily be formulated as an LCP which has always terminated in a complementary solution in numerical experiments, but has not yet been proven either the processability of MPG’s by Lemke’s algorithm or a counter example that it will not terminate with a solution. Till now, the processability of MPG’s by Lemke’s algorithm remains open. A general MPG (with arbitrary outgoing arcs) naturally reduces to a generalized linear complementarity problem (GLCP) involving a rectangular matrix where the vertices are represented by the columns and the outgoing arcs from each vertex are represented by rows in a particular block. The noteworthy result in this paper is that the GLCP obtained from an MPG is processable by Cottle–Dantzig principal pivoting algorithm which terminates with a solution. Several properties of the matrix which arise in this context are also discussed.

scholarly journals The Block Principal Pivoting Algorithm for the Linear Complementarity Problem with an M-Matrix

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Xi-Ming Fang ◽  
Zhi-Jun Qiao ◽  
Heng-Jun Zhao
Keyword(s):  
Linear Complementarity Problem ◽  
Lower Order ◽  
Complementarity Problem ◽  
Linear Complementarity ◽  
Higher Order ◽  
Direct Algorithm ◽  
Pivoting Algorithm ◽  
Principal Pivoting

The principal pivoting algorithm is a popular direct algorithm in solving the linear complementarity problem, and its block forms had also been studied by many authors. In this paper, relying on the characteristic of block principal pivotal transformations, a block principal pivoting algorithm is proposed for solving the linear complementarity problem with an M-matrix. By this algorithm, the linear complementarity problem can be solved in some block principal pivotal transformations. Besides, both the lower-order and the higher-order experiments are presented to show the effectiveness of this algorithm.

scholarly journals An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

2021 ◽  
Author(s):  
E. Alper Yıldırım
Keyword(s):  
Objective Function ◽  
Convex Relaxation ◽  
Large Family ◽  
Quadratic Program ◽  
Feasible Region ◽  
Recession Cone ◽  
Convex Relaxations ◽  
Quadratic Programs ◽  
Alternative Perspective ◽  
Optimal Value

AbstractWe study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

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