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. Second derivatives are assumed to be unavailable or too expensive to calculate.We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. The Hessian of the Lagrangian is approximated using a limited-memory quasi-Newton method.SNOPT is a particular implementation that uses a reduced-Hessian semidefinite QP solver (SQOPT) for the QP subproblems. It is designed for problems with many thousands of constraints and variables but is best suited for problems with a moderate number of degrees of freedom (say, up to 2000). Numerical results are given for most of the CUTEr and COPS test collections (about 1020 examples of all sizes up to 40000 constraints and variables, and up to 20000 degrees of freedom). become large), SNOPT enters nonlinear elastic mode and solves the problemis called a composite objective, and the penalty parameter γ (γ ≥ 0) may take a finite sequence of increasing values. If (NP) has a feasible solution and γ is sufficiently large, the solutions to (NP) and (NP(γ)) are identical. If (NP) has no feasible solution, (NP(γ)) will tend to determine a "good" infeasible point if γ is again sufficiently large. (If γ were infinite, the nonlinear constraint violations would be minimized subject to the linear constraints and bounds.) A similar 1 formulation of (NP) is used in the SQP method of Tone [98] and is fundamental to the S 1 QP algorithm of Fletcher [38]. See also Conn [25] and Spellucci [94]. An attractive feature is that only linear terms are added to (NP), giving no increase in the expected degrees of freedom at each QP solution. 1.2. The SQP Approach. An SQP method was first suggested by Wilson [102] for the special case of convex optimization. The approach was popularized mainly by Biggs [7], Han [66], and Powell [85, 87] for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in [61, 39, 73, 78, 28]. For a survey of recent results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solvers NLPQL [92], NPSOL [57, 60], and DONLP [95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers such as MINOS [75, 76, 77] and CONOPT [34, 2].SQP methods have been particularly successful in solving the optimization problems arising in optimal trajectory calculations. For many years, the optimal trajectory system OTIS (Hargraves and Paris [67]) has been...

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.We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. SNOPT is a particular implementation that makes use of a semidefinite QP solver. It is based on a limited-memory quasi-Newton approximation to the Hessian of the Lagrangian and uses a reduced-Hessian algorithm (SQOPT) for solving the QP subproblems. It is designed for problems with many thousands of constraints and variables but a moderate number of degrees of freedom (say, up to 2000). An important application is to trajectory optimization in the aerospace industry. Numerical results are given for most problems in the CUTE and COPS test collections (about 900 examples).

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Abstract.Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar's widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

This paper presents a fast algorithm for restoring video sequences. The proposed algorithm, as opposed to existing methods, does not consider video restoration as a sequence of image restoration problems. Rather, it treats a video sequence as a space-time volume and poses a space-time total variation regularization to enhance the smoothness of the solution. The optimization problem is solved by transforming the original unconstrained minimization problem to an equivalent constrained minimization problem. An augmented Lagrangian method is used to handle the constraints, and an alternating direction method is used to iteratively find solutions to the subproblems. The proposed algorithm has a wide range of applications, including video deblurring and denoising, video disparity refinement, and hot-air turbulence effect reduction.

Abstract. In recent years, several algorithms have appeared for modifying the factors of a matrix following a rank-one change. These methods have always been given in the context of specific applications and this has probably inhibited their use over a wider field. In this report, several methods are described for modifying Cholesky factors. Some of these have been published previously while others appear for the first time. In addition, a new algorithm is presented for modifying the complete orthogonal factorization of a general matrix, from which the conventional QR factors are obtained as a special case. A uniform notation has been used and emphasis has been placed on illustrating the similarity between different methods.

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