This paper proposes a new predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguished features of the method are full use of the BFGS quasi-Newton method in the corrector procedure and an application of the conjugate gradient method with an effective preconditioning matrix induced from the BFGS quasi-Newton method in the predictor procedure. Some preliminary numerical results are reported.

. This paper presents an algorithm for solving multi-stage stochastic convex nonlinear programs. The algorithm is based on the Lagrangian dual method which relaxes the nonanticipativity constraints, and the barrier function method which enhances the smoothness of the dual objective function so that the Newton search directions can be used. The algorithm is shown to be of global convergence and of polynomial-time complexity. Keywords: Multi-stage stochastic nonlinear programming, Lagrangian dual, Self-concordant barrier, Interior Point Methods, Polynomial-time Complexity. The research is partially supported by Grant RP972685 of National University of Singapore. 1 Introduction In this paper we propose an algorithm for multi-stage stochastic convex nonlinear programming (MSSCNP). In contrast with the two-stage SP, the multi-stage SP not only inherently has more scenarios but also more complicated structures of scenario trees. Therefore, the multi-stage SP is much more difficult to solve...

This paper proposes a new primal-dual predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguishing features of the method are full use of the BFGS quasi-Newton method in the corrector procedure and an application of the conjugate gradient method with an effective preconditioning matrix induced from the BFGS quasi-Newton method in the predictor procedure. Some preliminary numerical results are presented. Key words. Semidefinite Program, Linear Program over Symmetric Cones, SecondOrder Cone Program, Primal-Dual Interior-Point Method, Predictor-Corrector Method, Lagrangian Dual, Central Trajectory, BFGS Quasi-Newton Method, Conjugate Gradient Method, Conjugate Residual Method. 1 Introduction. Consider the equality standard form SDP and its dual Primal SDP: minimize A 0 ffl X subject to A p ffl X = b p (p = 1

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Multi-stage stochastic programming problems arise in many practical situations, such as production and manpower planning, portfolio selections and so on. Generally, the size of the deterministic equivalent of stochastic programs can be very large and not be solvable directly by optimization approaches. Sequential quadratic programming methods are iterative and very effective for solving medium-size nonlinear programming. Based on scenario analysis, a decomposition method based on SQP for solving a class of multistage nonlinear stochastic programs is proposed, which generates the search direction by solving parallelly a set of quadratic programming subproblems with size much less than the original problem at each iteration. Conjugate gradient methods can be introduced to derive the estimates of the dual multiplier associated with the nonanticipativity constraints. By selecting the step-size to reduce an exact penalty function sufficiently, the algorithm terminate finitely at an approxim...

We introduce two-stage stochastic semidefinite programs with recourse and present a Benders decomposition based linearly convergent interior point algorithms to solve them. This extends the results in Zhao [16] wherein it was shown that the logarithmic barrier associated with the recourse function of two-stage stochastic linear programs with recourse behaves as a strongly self-concordant barrier on the first stage solutions. In this paper we develop the necessary theory. A companion paper [8] addresses implementation issues for the theoretical algorithms of this paper.