Semidefinite Approaches for MIQCP: Convex Relaxations and Practical Methods

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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Convex Relaxations of Convolutional Neural Nets

ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) ◽

10.1109/icassp.2019.8683242 ◽

2019 ◽

Cited By ~ 1

Author(s):

Burak Bartan ◽

Mert Pilanci

Keyword(s):

Neural Nets ◽

Convex Relaxations

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Convex Relaxations for Permutation Problems

SIAM Journal on Matrix Analysis and Applications ◽

10.1137/130947362 ◽

2015 ◽

Vol 36 (4) ◽

pp. 1465-1488 ◽

Cited By ~ 7

Author(s):

F. Fogel ◽

R. Jenatton ◽

F. Bach ◽

A. d'Aspremont

Keyword(s):

Convex Relaxations ◽

Permutation Problems

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Elephant Herding Optimization for Energy-Based Localization

10.20944/preprints201807.0051.v1 ◽

2018 ◽

Cited By ~ 1

Author(s):

Sérgio Correia ◽

Marko Beko ◽

Luís Cruz ◽

Slavisa Tomic

Keyword(s):

Source Localization ◽

Wireless Sensors ◽

Performance Comparison ◽

Sensor Nodes ◽

Convex Relaxations ◽

Noisy Environments ◽

New Approach ◽

Metaheuristic Methods ◽

Simulation Results ◽

First Time

This work addresses the energy-based source localization problem in wireless sensors networks. Instead of circumventing the maximum likelihood (ML) problem by applying convex relaxations and approximations (like all existing approaches do), we here tackle it directly by the use of metaheuristics. To the best of our knowledge, this is the ﬁrst time that metaheuristics is applied to this type of problems. More speciﬁcally an elephant herding optimization (EHO) algorithm is applied. Through extensive simulations, the key parameters of the EHO algorithm are optimized such that they match the energy decay model between two sensor nodes. A detailed analysis of the computational complexity is presented, as well as performance comparison between the proposed algorithm and existing non-metaheuristic ones. Simulation results show that the new approach signiﬁcantly outperforms the existing solutions in noisy environments, encouraging further improvement and testing of metaheuristic methods.

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