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. 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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A simulation-management methodology is demonstrated for the rehabilitation of aquifers that have been subjected to chemical contamination. Finite element groundwater flow and contaminant transport simulation are combined with nonlinear optimization. The model is capable of determining well locations plus pumping and injection rates for groundwater quality control. Examples demonstrate linear or nonlinear objective functions subject to linear and nonlinear simulation and water management constraints. Restrictions can be placed on hydraulic heads, stresses, and gradients, in addition to contaminant concentrations and fluxes. These restrictions can be distributed over space and time. Three design strategies are demonstrated for an aquifer that is polluted by a constant contaminant source: they are pumping for contaminant removal, water injection for in-ground dilution, and a pumping, treatment, and injection cycle. A transient model designs either contaminant plume interception or in-ground dilution so that water quality standards are met. The method is not limited to these cases. It is generally applicable to the optimization of many types of distributed parameter systems. This paper is not subject to U.S. copyright. Published in 1984 by the American Geophysical Union.Paper number 3W1963. the three categories of alternatives for remedial action as physical containment, in situ aquifer rehabilitation, and withdrawal followed by treatment and use. Physical containment systems prevent the flow of contaminated groundwater with the aid of engineered features such as slurry trench cutoff walls, grout curtains, and hydrodynamic controls. Aquifer rehabilitation methods may involve injection and recharge systems that are supplemented by above-ground chemical-treatment or may involve injecting a neutralizing or biochemical degradation agent into the contaminated zone. The approach of withdrawal, treatment, and use does not attempt to exploit any assimilative capacity or attenuation properties of the aquifer and simply removes the contaminated water from the system. Chemical treatment processes and waste containment for aquifer decontamination have been discussed by Landon and SylVester [1982], Yaniga [1982] for hydrocarbon contamination, Stover [1982], McBride [1982], and Ohneck and Gardner [1982] for removal of toxic organic chemicals, Osiensky et al. [1982] for uranium tailings disposal problems, and Molsather and Barr [1982], as well as Giddings [1982] for landfill leachate containment. . Aquifer management research has also treated the problem of groundwater pollution control. Remson and Gorelick [1980]considered the optimal locations of pumping and injection wells and the determination of minimum pumping rates for the hydraulic containment of contaminated groundwater. Their approach was to include the finite difference approximation of the groundwater flow equation as constraints in a linear programing problem. Further constraints served to control hydraulic gradients which prevented the plume from spreading as well as ...

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