. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno-Todd-Ye type predictorcorrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno-Todd-Ye type predictor-corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, Infeasible-Interior-Point Method, Predictor-CorrectorMethod, Superlinear Convergence, Primal-Dual Nondegeneracy Abbreviated Title. Interior-Point Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japa...

This paper proposes a globally convergent predictor-corrector infeasible-interior-point algorithm for the monotone semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction, and shows its quadratic local convergence under the strict complementarity condition.

The Mizuno-Todd-Ye predictor-corrector algorithm for linear programming is extended for solving monotone linear complementarity problems from infeasible starting points. The proposed algorithm requires two matrix factorizations and at most three backsolves per iteration. Its computational complexity depends on the quality of the starting point. If the starting points are large enough then the algorithm has O(nL) iteration complexity. If a certain measure of feasibility at the starting point is small enough then the algorithm has O( p nL) iteration complexity. At each iteration both "feasibility' and "optimality" are reduced exactly at the same rate. The algorithm is quadratically convergent for problems having a strictly complementary solution, and therefore its asymptotic efficiency index is p 2. A variant of the algorithm can be used to detect whether solutions with norm less than a given constant exist. . Key Words:linear complementarity problems, predictor-corrector, infeasib...

Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primal-dual interior point method to solve the linear programming relaxations. A point which isagoodwarm start for a simplex-based cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some real-world problems; the algorithm appears to be competitive with a simplex-based cutting plane algorithm.

Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.

. The purpose of this technical report is twofold. The one is to present a globally convergent, predictor-corrector, primal-dual, infeasible-interior-point algorithm for SDPs (semidefinite programs). The algorithm is a special case of the generic interior-point algorithm (with a minor modification) proposed by Kojima, Shindoh and Hara [10] for SDLCPs (semidefinite linear complementarity problems). The other purpose is to study its local convergence; we show that a variation of the algorithm enjoys the superlinear convergence under a primal-dual nondegeneracy condition. Many interior-point algorithms [1, 3, 5, 6, 7, 16, 20, 21, etc.] have been already developed so far for SDPs, but the local convergence analysis has not been made for those algorithms. We place our main emphasis on the local convergence analysis. Quite recently, there has been made much progress in the primal-dual interior-point algorithms for SDPs. Monteiro [15] devised a new formulation of the primal-dual search direct...

A generalized class of infeasible-interior-point methods for solving horizontal linear complementarity problem is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced by methods in this class. In particular it is shown that the largest step path following algorithms generates convergent iterates even when starting from infeasible points. The computational complexity of the latter method is discussed in detail and its local convergent rate is analyzed. The primal-dual gap of the iterates produced by this method is superlinearly convergent to zero. A variant of the method has quadratic convergence.

We consider an infeasible-interior-point algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)). Keywords: Linear Programming, Average-Case Behavior, Infeasible-Interior-Point Algorithm. Running Title: Probabilistic Analysis of an LP Algorithm 1 Dept. of Management Sciences, University of Iowa. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 2 Dept. of Mathematics, Valdosta State University. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 3 Dept. of Mathematics, University of Iowa. Supported by ...

: A generalized class of infeasible-interior-point methods for solving horizontal linear complementarity problem is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced by methods in this class. In particular it is shown that the largest step path following algorithms generates convergent iterates even when starting from infeasible points. The computational complexity of the latter method is discussed in detail and its local convergent rate is analyzed. The primal-dual gap of the iterates produced by this method is superlinearly convergent to zero. A variant of the method has quadratic convergence. Key-words: Linear complementarity problem, infeasible central path, interior-point algorithm, predictorcorrector algorithm, largest step algorithm, shifted analytic center (R'esum'e : tsvp) INRIA, B.P. 105, 78153 Rocquencourt, France. Email: Frederic.Bonnans@inria.fr Department of Mathematics, The University of Iowa, Iowa City, Iowa 5224...

. Let M n (IK) denote the set of all n 2 n matrices with elements in IK, where IK represents the field IR of real numbers, the field 0 C of complex numbers or the (noncommutative) field IH of quaternion numbers. We call a subset T of M n (IK) a *-subalgebra of M n (IK) over the field IR (or simply a *-subalgebra) if (i) T forms a subring of M n (IK) with the usual addition A + B and multiplication AB of matrices A; B 2 M n (IK); specifically the zero matrix O and the identity matrix I belong to T . (ii) T is an IR-module, i.e., a vector space over the field IR; ffA + fiB 2 T for every ff; fi 2 IR and A; B 2 T , (iii) A 3 2 T if A 2 T , where A 3 denotes the conjugate transpose of A 2 M n (IK). The introduction of *-subalgebras T provides us with a unified and compact way of handling LPs (linear programs) in IR n , SDPs (semidefinite programs) in M n (IR), M n ( 0 C) and M n (IH), and monotone SDLCPs (semidefinite linear complementarity problems) in those spaces. We can extend t...