Performance of SSE and AVX Instruction Sets

SSE (streaming SIMD extensions) and AVX (advanced vector extensions) are SIMD (single instruction multiple data streams) instruction sets supported by recent CPUs manufactured in Intel and AMD. This SIMD programming allows parallel processing by multiple cores in a single CPU. Basic arithmetic and data transfer operations such as sum, multiplication and square root can be processed simultaneously. Although popular compilers such as GNU compilers and Intel compilers provide automatic SIMD optimization options, one can obtain better performance by a manual SIMD programming with proper optimization: data packing, data reuse and asynchronous data transfer. In particular, linear algebraic operations of vectors and matrices can be easily optimized by the SIMD programming. Typical calculations in lattice gauge theory are composed of linear algebraic operations of gauge link matrices and fermion vectors, and so can adopt the manual SIMD programming to improve the performance.Comment: 7 pages, 5 figures, 4 tables, Contribution to proceedings of the 30th International Symposium on Lattice Field Theory (Lattice 2012), June 24-29, 201

An exact duality theory for semidefinite programming based on sums of squares

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality is infeasible if and only if -1 lies in the quadratic module associated to it. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.Comment: arXiv admin note: substantial text overlap with arXiv:1108.593

Infeasibility certificates for linear matrix inequalities

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality is infeasible if and only if -1 lies in the quadratic module associated to it. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certificate, we use a more involved algebraic certificate motivated by the real radical and Prestel's theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination.Comment: 30 page

Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares

In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H_infinity control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series

Quantitative analysis of multi-periodic supply chain contracts with options via stochastic programming

We propose a stochastic programming approach for quantitative analysis of supply contracts, involving flexibility, between a buyer and a supplier, in a supply chain framework. Specifically, we consider the case of multi-periodic contracts in the face of correlated demands. To design such contracts, one has to estimate the savings or costs induced for both parties, as well as the optimal orders and commitments. We show how to model the stochastic process of the demand and the decision problem for both parties using the algebraic modeling language AMPL. The resulting linear programs are solved with a commercial linear programming solver; we compute the economic performance of these contracts, giving evidence that this methodology allows to gain insight into realistic problems.stochastic programming; supply contract; linear programming; modeling software; decision tree