Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compact Disjunctive Approximations to Nonconvex Quadratically
  Constrained Programs   Decades of advances in mixed-integer linear programming (MILP) and recent
development in mixed-integer second-order-cone programming (MISOCP) have
translated very mildly to progresses in global solving nonconvex mixed-integer
quadratically constrained programs (MIQCP). In this paper we propose a new
approach, namely Compact Disjunctive Approximation (CDA), to approximate
nonconvex MIQCP to arbitrary precision by \textit{convex} MIQCPs, which can be
solved by MISOCP solvers. For nonconvex MIQCP with $n$ variables and $m$
general quadratic constraints, our method yields relaxations with at most
$O(n\log(1/\varepsilon))$ number of continuous/binary variables and linear
constraints, together with $m$ \textit{convex} quadratic constraints, where
$\varepsilon$ is the approximation accuracy. The main novelty of our method
lies in a very compact lifted mixed-integer formulation for approximating the
(scalar) square function. This is derived by first embedding the square
function into the boundary of a three-dimensional second-order cone, and then
exploiting rotational symmetry in a similar way as in the construction of
BenTal-Nemirovski approximation. We further show that this lifted formulation
characterize the union of finite number of simple convex sets, which naturally
relax the square function in a piecewise manner with properly placed knots. We
implement (with JuMP) a simple adaptive refinement algorithm. Numerical
experiments on synthetic instances used in the literature show that our
prototypical implementation (with hundreds of lines of Julia code) can already
close a significant portion of gap left by various state-of-the-art global
solvers on more difficult instances, indicating strong promises of our proposed
approach.

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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph