Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underestimate the very global optima. The purpose of this article is to introduce an efficient and safe procedure to rigorously bound the global optima of semidefinite program. This work shows how, using interval arithmetic, rigorous error bounds for the optimal value can be computed by carefully post processing the output of a semidefinite programming solver. A lower bound is computed on a semidefinite relaxation of the constraint system and the objective function. Numerical results are presented using the SDPA (SemiDefinite Programming Algorithm), solver to compute the solution of semidefinite programs. This rigorous bound is injected in a branch and bound algorithm to solve the optimisation problem.
This paper provides a new variant of primal-dual interior-point method for solving a SemiDefinite Program (SDP). We use the PDIPA (primal-dual interior-point algorithm) solver entitled SDPA (SemiDefinite Programming Algorithm). This last uses a classical Newton descent method to compute the predictorcorrector search direction. The difficulty is in the computation of this line-search, it induces high computational costs. Here, instead we adopt a new procedure to implement another way to determine the step-size along the direction which is more efficient than classical line searches. This procedure consists in the computation of the step size in order to give a significant decrease along the descent line direction with a minimum cost. With this procedure we obtain à new variant of SDPA. The comparison of the results obtained with the classic SDPA and our new variant is promising.
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