A framework for quadratic form maximization over convex sets through nonconvex relaxations
A framework for quadratic form maximization over convex sets through nonconvex relaxations
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通过非凸松弛实现凸集二次形式最大化的框架
DOI：
10.1145/3406325.3451128
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Naor, Assaf
中科院分区：
文献类型：
--
作者：
Bhattiprolu, Vijay;Lee, Euiwoong;Naor, Assaf
我们研究以下优化问题的可逼近性。输入是ann×nmatrixA=（Aij），其中元素为真实的，并且是一个由成员预言机给出的原点对称凸体K_n。任务是计算（或近似）二次型∑i= 1 n ∑j= 1 nAijxixj =<$x，Ax <$asx在K上的最大值。这是一个丰富而富有表现力的优化问题家族;对于不同的矩阵A和凸体选择，套件包括各种优化问题，如最大割，Grothendieck/非交换Grothendieck不等式，小集合扩展等。虽然文献研究了这些特殊情况下使用的情况下，具体的推理，在这里，我们开发了一个通用的方法来治疗的逼近性和不可逼近性方面的这些问题。潜在的几何ofK起着至关重要的作用，我们表明，在常用的复杂性假设下，多时间常数近似性需要thatK有2型常数，增长缓慢，随着n。然而，我们表明，即使当2型常数是有界的，这个问题有时表现出很强的近似硬度。因此，即使在2型机构的领域，逼近景观是微妙的和微妙的。然而，我们建立优化和几何之间的联系，使我们能够设计一个通用的算法方法，上述问题。我们关联到每个凸体一个新的（高维）辅助集，是不凸的，但近似凸时K有界的2型常数。如果我们的辅助集有一个近似的分离预言，那么我们设计一个近似算法的原始二次优化问题，使用近似版本的椭球方法。尽管我们的硬度结果意味着这样的预言一般不存在，但这个新问题可以在特定的感兴趣的情况下通过实现一系列来自泛函分析的经典工具来解决，最值得注意的是线性算子的深度因子分解理论。除了包含文献中发现的常数因子近似算法的场景之外，我们的通用框架意味着，对于具有有界2型常数的凸集，常数因子逼近性在以下基本运算下保持不变：（a）子空间，（B）商，（c）Minkowski和，（d）复插值。这产生了一个丰富的家庭的新的例子，常数因子近似是可能的，这是超越了以前的方法。我们还表明（在常用的复杂性假设下），对于对称范数和酉不变矩阵范数，2型常数几乎表征了二次最大化的可逼近性。
We investigate the approximability of the following optimization problem. The input is ann×nmatrixA=(Aij) with real entries and an origin-symmetric convex bodyK⊂ ℝnthat is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑i=1n∑j=1nAijxixj=⟨x,Ax⟩ asxranges overK. This is a rich and expressive family of optimization problems; for different choices of matricesAand convex bodiesKit includes a diverse range of optimization problems like max-cut, Grothendieck/non-commutative Grothendieck inequalities, small set expansion and more. While the literature studied these special cases using case-specific reasoning, here we develop a general methodology for treatment of the approximability and inapproximability aspects of these questions.The underlying geometry ofKplays a critical role; we show under commonly used complexity assumptions that polytime constant-approximability necessitates thatKhas type-2 constant that grows slowly withn. However, we show that even when the type-2 constant is bounded, this problem sometimes exhibits strong hardness of approximation. Thus, even within the realm of type-2 bodies, the approximability landscape is nuanced and subtle.However, the link that we establish between optimization and geometry of Banach spaces allows us to devise a generic algorithmic approach to the above problem. We associate to each convex body a new (higher dimensional) auxiliary set that is not convex, but is approximately convex whenKhas a bounded type-2 constant. If our auxiliary set has an approximate separation oracle, then we design an approximation algorithm for the original quadratic optimization problem, using an approximate version of the ellipsoid method. Even though our hardness result implies that such an oracle does not exist in general, this new question can be solved in specific cases of interest by implementing a range of classical tools from functional analysis, most notably the deep factorization theory of linear operators.Beyond encompassing the scenarios in the literature for which constant-factor approximation algorithms were found, our generic framework implies that that for convex sets with bounded type-2 constant, constant factor approximability is preserved under the following basic operations: (a) Subspaces, (b) Quotients, (c) Minkowski Sums, (d) Complex Interpolation. This yields a rich family of new examples where constant factor approximations are possible, which were beyond the reach of previous methods. We also show (under commonly used complexity assumptions) that for symmetric norms and unitarily invariant matrix norms the type-2 constant nearly characterizes the approximability of quadratic maximization.
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