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

Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques for solving the subproblems occurring in the algorithm. Second derivative information can be used, but when it is not available, limited memory quasi-Newton approximations are made. The performance of the code is studied using a set of difficult test problems from the CUTE collection.

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Mathematical programs with nonlinear complementarity constraints are reformulated using better-posed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations. We present two globally convergent algorithms based on sequential quadratic programming, SQP, as applied in exact penalty methods for nonlinear programs. Global convergence of the implicit smooth SQP method depends on existence of a lower-level nondegenerate (strictly complementary) limit point of the iteration sequence. Global convergence of the explicit smooth SQP method depends on a weaker property, i.e. existence of a limit point at which a generalized constraint qualification holds. We also discuss some practical matter...

Direct boundary value problem methods in combination with SQP iteration have proved to be very successful in solving nonlinear optimal control problems. Such methods use parameterized control functions, discretize the state differential equations by, e.g., multiple shooting or collocation, and treat the discretized boundary value problem as an equality constraint in a large, nonlinear, constrained optimization problem. In real-life applications several thousand variables may appear in the NLP. Solution by standard techniques is therefore impractical. This dissertation develops a general concept for a class of structured direct SQP methods based on a decoupling strategy. A careful choice of the discretization reveals an inherent multistage block structure of the QP subproblems. We present a recursive solution algorithm for the associated KKT systems which makes full use of this sparse structure, and propose a structure-preserving primal-dual interior point method for treating the genera...

. A sequential quadratic programming (SQP) algorithm generating feasible iterates is described and analyzed. What distinguishes this algorithm from previous feasible SQP algorithms proposed by various authors is a reduction in the amount of computation required to generate a new iterate while the proposed scheme still enjoys the same global and fast local convergence properties. A preliminary implementation has been tested and some promising numerical results are reported. Key words. sequential quadratic programming, SQP, feasible iterates, feasible SQP, FSQP AMS subject classifications. 49M37, 65K05, 65K10, 90C30, 90C53 PII. S1052623498344562 1.

. At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which @f(p) has 0-breadth). This property opens the way to defining a second derivative of f at p, along U . We do this via an intermediate function, convex on U . We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization. 1. Introduction This paper deals with higher-order expansions of a nonsmooth function, a problem addressed in [4], [5], [7], [13], [29] among others. The initial motivation for our present work lies in the following facts. When trying to generalize the classical second-order Taylor expansion of a function f at a nondifferentiability point p, the major difficulty is by far the nonlinear...

. 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...

. It was recently shown that, in the solution of smooth constrained optimization problems by sequential quadratic programming (SQP), the Maratos effect can be prevented by means of a certain nonmonotone (more precisely, three-step or four-step monotone) line search. Using a well known transformation, this scheme can be readily extended to the case of minimax problems. It turns out however that, due to the structure of these problems, one can use a simpler scheme. Such a scheme is proposed and analyzed in this paper. Numerical experiments indicate a significant advantage of the proposed line search over the (monotone) Armijo search. Key words. Minimax problems, SQP direction, Maratos effect, Superlinear convergence. 1 This research was supported in part by NSF's Engineering Research Centers Program No. NSFD-CDR88 -03012, by NSF grant No. DMC-88-15996 and by a grant from the Westinghouse Corporation. 2 To whom the correspondence should be addressed. 1. Introduction. Consider the "m...