Reduction of indefinite quadratic programs to bilinear programs

1992 ◽  
Vol 2 (1) ◽  
pp. 41-60 ◽  
Author(s):  
Pierre Hansen ◽  
Brigitte Jaumard
Keyword(s):  
Quadratic Programs ◽  
Indefinite Quadratic ◽  
Bilinear Programs

A Sparsity Preserving Convexification Procedure for Indefinite Quadratic Programs Arising in Direct Optimal Control

2017 ◽  
Vol 27 (3) ◽  
pp. 2085-2109 ◽  
Author(s):  
Robin Verschueren ◽  
Mario Zanon ◽  
Rien Quirynen ◽  
Moritz Diehl
Keyword(s):  
Optimal Control ◽  
Quadratic Programs ◽  
Direct Optimal Control ◽  
Indefinite Quadratic

scholarly journals The Equivalence of Quadratic Forms

1957 ◽  
Vol 9 ◽  
pp. 526-548 ◽  
Author(s):  
G. L. Watson
Keyword(s):  
Quadratic Forms ◽  
Indefinite Quadratic ◽  
Number Of Classes

The main object of this paper is to find the number of classes in a genus of indefinite quadratic forms, with integral coefficients, in k ≥ 4 variables, distinguishing for even k two cases, according as improper equivalence is or is not admitted.

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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