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A globally convergent algorithm based on the stabilized sequential quadratic programming (sSQP) method is presented in order to solve optimization problems with equality constraints and bounds. This formulation has attractive features in the sense that constraint qualifications are not needed at all. In contrast with classic globalization strategies for Newton-like methods, we do not make use of merit functions. Our scheme is based on performing corrections on the solutions of the subproblems by using an inexact restoration procedure. The presented method is well defined and any accumulation point of the generated primal sequence is either a Karush-Kuhn-Tucker point or a stationary (maybe feasible) point of the problem of minimizing the infeasibility. Also, under suitable hypotheses, the sequence generated by the algorithm converges Q-linearly. Numerical experiments are given to confirm theoretical results.

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Linearly constrained optimization problems with simple bounds are considered in the present work. First, a preconditioned spectral gradient method is defined for the case in which no simple bounds are present. This algorithm can be viewed as a quasiNewton method in which the approximate Hessians satisfy a weak secant equation. The spectral choice of steplength is embedded into the Hessian approximation, and the whole process is combined with a nonmonotone line search strategy. The simple bounds are then taken into account by placing them in an exponential penalty term that modifies the objective function. The exponential penalty scheme defines the outer iterations of the process. Each outer iteration involves the application of the previously defined preconditioned spectral gradient method for linear equality constrained problems. Therefore, an equality constrained convex quadratic programming problem needs to be solved at every inner iteration. The associated extended KKT matrix remains constant unless the process is reinitiated. In ordinary inner iterations, only the right hand side of the KKT system changes. Therefore, suitable sparse factorization techniques can be effectively applied and exploited. Encouraging numerical experiments are presented.

Inexact restoration methods, Filter methods, Merit function, Hard-spheres problems,