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Grothendieck’s inequality [Grothendieck, 1953] states that there is an absolute constant K > 1 such that for any n× n matrix A, ‖A‖_{∞→1} := max_{s,t ∈ {± 1}ⁿ}∑_{i,j} A[i,j]⋅s(i)⋅t(j) ≥ 1/K ⋅ max_{u_i,v_j ∈ S^{n-1}}∑_{i,j} A[i,j]⋅⟨u_i,v_j⟩. In addition to having a tremendous impact on Banach space theory, this inequality has found applications in several unrelated fields like quantum information, regularity partitioning, communication complexity, etc. Let K_G (known as Grothendieck’s constant) denote the smallest constant K above. Grothendieck’s inequality implies that a natural semidefinite programming relaxation obtains a constant factor approximation to ‖A‖_{∞ → 1}. The exact value of K_G is yet unknown with the best lower bound (1.67…) being due to Reeds and the best upper bound (1.78…) being due to Braverman, Makarychev, Makarychev and Naor [Braverman et al., 2013]. In contrast, the little Grothendieck inequality states that under the assumption that A is PSD the constant K above can be improved to π/2 and moreover this is tight. The inapproximability of ‖A‖_{∞ → 1} has been studied in several papers culminating in a tight UGC-based hardness result due to Raghavendra and Steurer (remarkably they achieve this without knowing the value of K_G). Briet, Regev and Saket [Briët et al., 2015] proved tight NP-hardness of approximating the little Grothendieck problem within π/2, based on a framework by Guruswami, Raghavendra, Saket and Wu [Guruswami et al., 2016] for bypassing UGC for geometric problems. This also remained the best known NP-hardness for the general Grothendieck problem due to the nature of the Guruswami et al. framework, which utilized a projection operator onto the degree-1 Fourier coefficients of long code encodings, which naturally yielded a PSD matrix A. We show how to extend the above framework to go beyond the degree-1 Fourier coefficients, using the global structure of optimal solutions to the Grothendieck problem. As a result, we obtain a separation between the NP-hardness results for the two problems, obtaining an inapproximability result for the Grothendieck problem, of a factor π/2 + ε₀ for a fixed constant ε₀ > 0. 

We investigate the approximability of the following optimization problem. The input is an n× n matrix A=(Aij) with real entries and an origin-symmetric convex body K⊂ ℝn that is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑i=1n∑j=1n Aij xixj=⟨ x,Ax⟩ as x ranges over K. This is a rich and expressive family of optimization problems; for different choices of matrices A and convex bodies K it 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 of K plays a critical role; we show under commonly used complexity assumptions that polytime constant-approximability necessitates that K has type-2 constant that grows slowly with n. 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 when K has 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.