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

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  • E. Alper Yıldırım

    (The University of Edinburgh)

Abstract

We 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.

Suggested Citation

  • E. Alper Yıldırım, 2022. "An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs," Journal of Global Optimization, Springer, vol. 82(1), pages 1-20, January.
    • Handle: RePEc:spr:jglopt:v:82:y:2022:i:1:d:10.1007_s10898-021-01066-3
    DOI: 10.1007/s10898-021-01066-3
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    References listed on IDEAS

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    1. João Gouveia & Ting Kei Pong & Mina Saee, 2020. "Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices," Journal of Global Optimization, Springer, vol. 76(2), pages 383-405, February.
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    5. Sunyoung Kim & Masakazu Kojima & Kim-Chuan Toh, 2020. "Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures," Journal of Global Optimization, Springer, vol. 77(3), pages 513-541, July.
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