Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this articl...
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Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this article Bell inequalities and their violations are considered through the lens of noncommutative polynomial optimization. Optimality of these violations is certified for a large majority of a set of standard Bell inequalities, denoted A2--A89 in the literature. The main techniques used in the paper include the NPA hierarchy, i.e., the noncommutative version of the Lasserre semidefinite programming (SDP) hierarchies based on the Helton-McCullough Positivstellensatz, the Gelfand--Naimark--Segal (GNS) construction with a novel use of the Artin-Wedderburn theory for rounding and projecting, and nonlinear programming (NLP). A new ``Newton chip"-like technique for reducing sizes of SDPs arising in the constructed polynomial optimization problems is presented. This technique is based on conditional expectations. Finally, noncommutative Gro"\bner bases are exploited to certify when an optimizer (a solution yielding optimum violation) cannot be extracted from a dual SDP solution.
The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, le...
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The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, l-dimensional kernel of the constraint matrix, by the linear independence of a set of l(l + 1)/2 derivative vectors. We show that this linear independence requirement can be equivalently formulated in a smaller set, of l derivative vectors, by considering all orthonormal bases of the kernel instead. This allows us to identify that not all bases are relevant for a constraint qualification to be defined, giving rise to a strictly weaker variant of nondegeneracy related to the global convergence of an external penalty method. We use some of these ideas to revisit an approach of Forsgren (Math Program 88, 105-128, 2000) for exploiting the sparsity structure of a transformation of the constraints to define a constraint qualification, which led us to develop another relaxed notion of nondegeneracy using a simpler transformation. If the zeros of the derivatives of the constraint function at a given point are considered, instead of the zeros of the function itself in a neighborhood of that point, we obtain an even weaker constraint qualification that connects Forsgren's condition and ours.
We consider beamformer design for multiuser multiple-input multiple-output (MIMO) interference channels where each transmitter communicates to its intended receivers while sharing the same spectro-temporal resources w...
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We consider beamformer design for multiuser multiple-input multiple-output (MIMO) interference channels where each transmitter communicates to its intended receivers while sharing the same spectro-temporal resources with some other unintended receivers. This scenario is often referred to as MIMO interference broadcast channel (IBC) where by using interference alignment (IA), it is possible to cancel out inter and intracell interferences so that the achievable degrees of freedom could be linearly scaled up with the number of users. In this paper, we propose two distinct and novel IA algorithms, one of which is IA via signal matching (SIGMA) that revolves around interference leakage minimization and further takes the desired signal subspaces into account to output higher sum rates. The second algorithm is called bipartite rank fitting, which is founded on the prior knowledge of the rank of receive beamformers and interference matrices. Therefore, this algorithm is quite a unique approach towards interference alignment in MIMO IBCs since it performs the optimization over only receive beamformers. Furthermore, we propose a robust design of SIGMA algorithm to achieve better performance in the presence of imperfect channel state information (CSI). The key tenet of our design is that we formulate the optimization problems involved in these two algorithms as plain semidefinite programs. It is shown that the proposed algorithms are able to outperform state-of-the-art IA techniques under perfect and imperfect CSI, and the suitability of each algorithm for different network configurations is further discussed.
A proximal augmented method for solving semidefinite programs is introduced. This method is based on the Augmented Lagrangian method. We study the theoretical properties and show the convergence of the new approach un...
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A proximal augmented method for solving semidefinite programs is introduced. This method is based on the Augmented Lagrangian method. We study the theoretical properties and show the convergence of the new approach under weaker conditions. Numerical results on some semidefinite programming problems are reported and show that the new method is potentially efficient. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.
We propose a framework for developing and analyzing primal-dual interior point algorithms for semidefinite programming. This framework is an extension of the v-space approach that was developed by Kojima et al. (1991)...
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We propose a framework for developing and analyzing primal-dual interior point algorithms for semidefinite programming. This framework is an extension of the v-space approach that was developed by Kojima et al. (1991) for linear complementarity problems. The extension to semidefinite programming allows us to interpret Nesterov-Todd type directions (Nesterov and Todd 1995, 1997) as Newton search directions. Our approach does not involve any barrier function. Several primal-dual path-following algorithms for semidefinite programming are analyzed. The treatment of these algorithms for semidefinite programming in our setting bears great similarity to the linear programming case. (C) 1999 Published by Elsevier Science B.V. and IMACS. All rights reserved.
The purpose of this paper is to develop certain geometric results concerning the feasible regions of semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. ...
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The purpose of this paper is to develop certain geometric results concerning the feasible regions of semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. More specifically, given a point x in a spectrahedron, we derive an expression for the minimal face containing x. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, ...
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Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance, for FPGAs or custom hardware implementations. This problem becomes challenging when the program does not employ solely linear operations as non-linearities are inherent to many interesting computational problems in real-world applications. Existing solutions to reasoning possibly lead to either inaccurate bounds or high analysis time in the presence of nonlinear correlations between variables. Furthermore, while it is easy to implement a straightforward method such as interval arithmetic, sophisticated techniques are less straightforward to implement in a formal setting. Thus there is a need for methods that output certificates that can be formally validated inside a proof assistant. We present a framework to provide upper bounds on absolute roundoff errors of floating-point nonlinear programs. This framework is based on optimization techniques employing semidefinite programming and sums of squares certificates, which can be checked inside the Coq theorem prover to provide formal roundoff error bounds for polynomial programs. Our tool covers a wide range of nonlinear programs, including polynomials and transcendental operations as well as conditional statements. We illustrate the efficiency and precision of this tool on non-trivial programs coming from biology, optimization, and space control. Our tool produces more accurate error bounds for 23% of all programs and yields better performance in 66% of all programs.
A long-standing problem related to floating-point implementation of numerical programs is to provide efficient yet precise analysis of output errors. We present a framework to compute lower bounds on largest absolute ...
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A long-standing problem related to floating-point implementation of numerical programs is to provide efficient yet precise analysis of output errors. We present a framework to compute lower bounds on largest absolute roundoff errors, for a particular rounding model. This method applies to numerical programs implementing polynomial functions with box constrained input variables. Our study is based on three different hierarchies, relying respectively on generalized eigenvalue problems, elementary computations, and semidefinite programming (SDP) relaxations. This is complementary of over-approximation frameworks, consisting of obtaining upper bounds on the largest absolute roundoff error. Combining the results of both frameworks allows one to get enclosures for upper bounds on roundoff errors. The under-approximation framework provided by the third hierarchy is based on a new sequence of convergent robust SDP approximations for certain classes of polynomial optimization problems. Each problem in this hierarchy can be solved exactly via SDP. By using this hierarchy, one can provide a monotone non-decreasing sequence of lower bounds converging to the absolute roundoff error of a program implementing a polynomial function, applying for a particular rounding model. We investigate the efficiency and precision of our method on nontrivial polynomial programs coming from space control, optimization, and computational biology.
We use rank one Gaussian perturbations to derive a smooth stochastic approximation of the maximum eigenvalue function. We then combine this smoothing result with an optimal smooth stochastic optimization algorithm to ...
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We use rank one Gaussian perturbations to derive a smooth stochastic approximation of the maximum eigenvalue function. We then combine this smoothing result with an optimal smooth stochastic optimization algorithm to produce an efficient method for solving maximum eigenvalue minimization problems, and detail a variant of this stochastic algorithm with monotonic line search. Overall, compared to classical smooth algorithms, this method runs a larger number of significantly cheaper iterations and, in certain precision/dimension regimes, its total complexity is lower than that of deterministic smoothing algorithms.
We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton's method for self-concordant functions, including th...
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We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton's method for self-concordant functions, including the case of inexact search directions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori and Teboulle [it Math. Program., 145 (2014), pp. 451-482], and extends recent performance estimation results for the method of Cauchy by the authors [it Optim. Lett., 11 (2017), pp. 1185-1199]. To illustrate the applicability of the tools, we demonstrate a novel complexity analysis of short step interior point methods using inexact search directions. As an example in this framework, we sketch how to give a rigorous worst-case complexity analysis of a recent interior point method by Abernethy and Hazan [it PMLR, 48 (2016), pp. 2520-2528].
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