Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available, and that the constraint gradients are sparse.

this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can

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. In this paper some Newton and quasi-Newton algorithms for the solution of inequality constrained minimization problems are considered. All the algorithms described produce sequences fx k g converging q-superlinearly to the solution. Furthermore, under mild assumptions, a q-quadratic convergence rate in x is also attained. Other features of these algorithms are that the solution of linear systems of equations only is required at each iteration and that the strict complementarity assumption is never invoked. First the superlinear or quadratic convergence rate of a Newton-like algorithm is proved. Then, a simpler version of this algorithm is studied and it is shown that it is superlinearly convergent. Finally, quasi-Newton versions of the previous algorithms are considered and, provided the sequence defined by the algorithms converges, a characterization of superlinear convergence extending the result of Boggs, Tolle and Wang is given. Key Words. Inequality constrained optimization, New...

. In this paper we give a global convergence analysis of a basic version of an SQP algorithm described in [2] for the solution of large scale nonlinear inequality-constrained optimization problems. Several procedures and options have been added to the basic algorithm to improve the practical performance; some of these are also analyzed. The important features of the algorithm include the use of a constrained merit function to assess the progress of the iterates and a sequence of approximate merit functions that are less expensive to evaluate. It also employs an interior point quadratic programming solver that can be terminated early to produce a truncated step. Key words. Sequential Quadratic Programming, Global Convergence, Merit Function, Large Scale Problems. AMS subject classifications. 49M37, 65K05, 90C30 1. Introduction. In this report we consider an algorithm to solve the inequalityconstrained minimization problem, min x f(x) subject to: g(x) 0; (1.1) where x 2 R n , and...

The constrained least-squares regularization of nonlinear ill-posed problems is a nonlinear programming problem for which trust-region methods have been developed. In this paper the convergence theory of one of those methods is addressed. It will be proved that, under suitable hypotheses, local (superlinear or quadratic) convergence holds and every accumulation point is second-order stationary. Key words. Trust-region methods, Regularization, Ill Conditioning, Ill-Posed Problems, Constrained Minimization, Fixed-Point QuasiNewton methods. 1 Introduction Many practical problems in applied sciences and engineering give rise to ill-conditioned (linear or nonlinear) systems F (x) = y (1) where F : IR n ! IR m . Neither "exact solutions" of (1) (when they exist), nor global minimizers of kF (x) \Gamma yk have physical meaning since they are, to a great extent, contaminated by the influence of measuring and rounding errors and, perhaps, uncertainty in the model formulation. From the ...

An algorithm for general nonlinearly constrained optimization is presented, which solves an unconstrained piecewise quadratic subproblem and a quadratic programming subproblem at each iterate. The algorithm is robust since it can circumvent the difficulties associated with the possible inconsistency of QP subproblem of the original SQP method. Moreover, the algorithm can converge to a point which satisfies a certain first-order necessary optimality condition even when the original problem is itself infeasible, which is a feature of Burke and Han's methods(1989). Unlike Burke and Han's methods(1989), however, we do not introduce additional bound constraints. The algorithm solves the same subproblems as Han-Powell SQP algorithm at feasible points of the original problem. Under certain assumptions, it is shown that the algorithm coincide with the Han-Powell method when the iterates are sufficiently close to the solution. Some global convergence results are proved and local superlinear co...

As is well known, superlinear or quadratic convergence of the primal-dual sequence generated by an optimization algorithm does not, in general, imply superlinear convergence of the primal part. Primal convergence, however, is often of particular interest. For the sequential quadratic programming (SQP) algorithm, local primal-dual quadratic convergence can be established under the assumptions of uniqueness of the Lagrange multiplier associated to the solution and the second-order sufficient condition. At the same time, previous primal superlinear convergence results for SQP required to strengthen the first assumption to the linear independence constraint qualification. In this paper, we show that this strengthening of assumptions is actually not necessary. Specifically, we show that once primal-dual convergence is assumed or already established, for primal superlinear rate one only needs a certain error bound estimate. This error bound holds, for example, under the second-order sufficient condition, which is needed for primal-dual local analysis in any case. Moreover, in some situations even second-order sufficiency can be relaxed to the weaker assumption that the multiplier in question is noncritical. Our study is performed for a rather general perturbed SQP framework, which covers in addition to SQP and quasi-Newton SQP some other algorithms as well. For example, as a by-product,

The sequential quadratic programming (SQP) algorithm has been one of the most successful general methods for solving nonlinear constrained optimization problems. We provide an introduction to the general method and show its relationship to recent developments in interior-point approaches. We emphasize large-scale aspects. Key words: sequential quadratic programming, nonlinear optimization, Newton methods, interior-point methods, local convergence, global convergence ? Contribution of Sandia National Laboratories and not subject to copyright in the United States. Preprint submitted to Elsevier Preprint 1 July 1999 1 Introduction In this article we consider the general method of Sequential Quadratic Programming (hereafter denoted SQP) for solving the nonlinear programming problem minimize f(x) x subject to: h(x) = 0 g(x) 0 (NLP) where f : R n ! R, h : R n ! R m , and g : R n ! R p . Broadly defined, the SQP method is a procedure that generates iterates converging ...

In this paper, we present a BFGS method for solving a KKT system in mathematical programming, based on a nonsmooth equation reformulation of the KKT system. We successively split the nonsmooth equation into equivalent equations with particular structure. Based on the splitting, we develop a BFGS method in which subproblems are systems of linear equations with symmetric and positive definite coefficient matrices. A suitable line search is introduced under which the generated iterates exhibit an approximately norm decent property. The method is well defined and, under suitable conditions, converges to a KKT point globally and superlinearly without convexity assumption on the problem.