Mathematical Sciences Research Institute

About: Mathematical Sciences Research Institute is a nonprofit organization based out in Berkeley, California, United States. It is known for research contribution in the topics: Boundary (topology) & Cohomology. The organization has 552 authors who have published 881 publications receiving 42002 citations. The organization is also known as: MSRI.

On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators

Abstract: This paper shows, by means of an operator called asplitting operator, that the Douglas--Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas--Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.

Greedy Randomized Adaptive Search Procedures

Abstract: Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.

The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes

Abstract: Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z/sub 4/, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z/sub 4/ domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z/sub 4/-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z/sub 4/-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z/sub 4/, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the I(Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z/sub 4/, but extended Hamming codes of length n/spl ges/32 and the Golay code are not. Using Z/sub 4/-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code. >

A new polynomial invariant of knots and links

Abstract: The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation

Most Tensor Problems Are NP-Hard

Abstract: We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.