Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J. Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximities for linear optimization (LO) problems. They also extended the approach for LO to SDO. In this paper we present a primal-dual interior-point algorithm for SDO problems based on a simple kernel function which was first presented at the Proceedings of Industrial Symposium and Optimization Day, Australia, November 2002; the function is not self-regular. We derive the complexity analysis for algorithms based on this kernel function, both with large- and small-updates. The complexity bounds are and , respectively, which are as good as those in the linear case. Mathematics Subject Classifications (2000) 90C22, 90C31.     

Counting and enumeration complexity with application to multicriteria scheduling

In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results.Received: November 2004 / Received version: March 2005MSC classification: 90B40, 90C29, 68Q15     

On the Computation of Square Roots in Finite Fields

In this paper, two improvements for computing square roots in finite fields are presented. Firstly, we give a simple extension of a method by O. Atkin, which requires two exponentiations in FM _q , when q9 mod 16. Our second method gives a major improvement to the Cipolla–Lehmer algorithm, which is both easier to implement and also much faster. While our method is independent of the power of 2 in q–1, its expected running time is equivalent to 1.33 as many multiplications as exponentiation via square and multiply. Several numerical examples are given that show the speed-up of the proposed methods, compared to the routines employed by Mathematica, Maple, respectively Magma.     

On the quantifier complexity of Δ<Subscript> n+1</Subscript> (T)– induction

In this paper we continue the study of the theories I _n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that I _n+1 (T) is _n+2 –axiomatizable. In particular, I _n+1 (I _n+1 ) gives an axiomatization of Th _n+2 (I _n+1 ) and is not finitely axiomatizable. This fact relates the fragment I _n+1 (I _n+1 ) to induction rule for _n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for _n+2 and _n+1 formulas (see [2]).Research partially supported by grant PB96–1345 (Spanish Goverment)Mathematics Subject Classification (2000): 03F30, 03H15     

Network synthesis under survivability constraints

Telecommunication networks are subject to link and equipment failures. Since failures cannot be entirely avoided, networks have to be designed so as to survive failure situations. In this paper, we are interested in designing low cost survivable networks. Given point-to-point traffic demands and a cost/capacity function for each link, we aim at finding the minimum cost capacities satisfying the given demands and survivability requirements. A survivability model that reroutes interrupted traffic using all the available capacities on the network is presented and studied. In the proposed model, capacity and flow assignments for each network operating state are jointly optimized. We prove the -hardness of the optimisation problem defined by dual constraints. Then, we propose a polynomial relaxation along with a fast heuristic to compute a feasible solution of the problem from its relaxed optimal solution. Our solution approaches are tested on a set of problem instances.Received: September 2002, Revised: July 2003, AMS classification: 90C05     

On the Ideals of Secant Varieties of Segre Varieties

We establish basic techniques for determining the ideals of secant varieties of Segre varieties.We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.     

Studying the Complexity of Global Verification for NP-Hard Discrete Optimization Problems

This paper examines the complexity of global verification for MAX-SAT, MAX-k-SAT (for k3), Vertex Cover, and Traveling Salesman Problem. These results are obtained by adaptations of the transformations that prove such problems to be NP-complete. The class of problems PGS is defined to be those discrete optimization problems for which there exists a polynomial time algorithm such that given any solution , either a solution can be found with a better objective function value or it can be concluded that no such solution exists and is a global optimum. This paper demonstrates that if any one of MAX-SAT, MAX-k-SAT (for k3), Vertex Cover, or Traveling Salesman Problem are in PGS, then P=NP.     

Irreducibility testing over local fields

The purpose of this paper is to describe a method to determine whether a bivariate polynomial with rational coefficients is irreducible when regarded as an element in , the ring of polynomials with coefficients from the field of Laurent series in with rational coefficients. This is achieved by computing certain associated Puiseux expansions, and as a result, a polynomial-time complexity bound for the number of bit operations required to perform this irreducibility test is computed.

     

Bistable behaviour in squeezed vacua: II. Stability analysis and chaos

Linear stability analysis and (numerical) investigation of the periodic and chaotic self-pulsing behaviour are presented for the Maxwell-Bloch equations of a bistable model in contact with a squeezed vacuum field. Effect of the squeeze phase parameter on the period doubling bifurcation that preceeds chaos is examined for the adiabatic and non-adiabatic regimes.