Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

Nonconvex optimization problems with an L1constraint are ubiquitous, and are found in many application domains including: optimal control of hybrid systems, machine learning and statistics, and operations research. This paper shows that nonconvex optimization problems with an L1constraint can be approximately solved in polynomial time. We first show that nonlinear integer programs with an L1constraint can be solved in a number of oracle steps that is polynomial in the dimension of the decision variable, for each fixed radius of the L1-constraint. When specialized to polynomial integer programs, our result shows that such problems have a time complexity that is polynomial in simultaneously both the dimension of the decision variables and number of constraints, for each fixed radius of the L1-constraint. We prove this result using a geometric argument that leverages ideas from stochastic process theory and from the theory of convex bodies in high-dimensional spaces. We conclude by providing an additive polynomial time approximation scheme (PTAS) for continuous optimization of Lipschitz functions subject to Lipschitz constraints intersected with an L1-constraint, and we sketch a generalization to mixed-integer optimization.

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An FPTAS for Minimizing Indefinite Quadratic Forms over Integers in Polyhedra

Hildebrand¹,

Weismantel²,

Zemmer³

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We present a generic approach that allows us to develop a fully polynomial-time approximation scheme (FTPAS) for minimizing nonlinear functions over the integer points in a rational polyhedron in fixed dimension. The approach combines the subdivision strategy of Papadimitriou and Yannakakis [22] with ideas similar to those commonly used to derive real algebraic certificates of positivity for polynomials. Our general approach is widely applicable. We apply it, for instance, to the Motzkin polynomial and to indefinite quadratic forms x T Qx in a fixed number of variables, where Q has at most one positive, or at most one negative eigenvalue. In dimension three, this leads to an FPTAS for general Q.

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Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

Cited by 6 publications

References 19 publications

Complexity of optimizing over the integers

Complexity of optimizing over the integers

Polynomial-time approximation for nonconvex optimization problems with an L1-constraint

An FPTAS for Minimizing Indefinite Quadratic Forms over Integers in Polyhedra

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