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SNOPT: an SQP algorithm for large-scale constrained optimization. (English) Zbl 1210.90176

Summary: 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).

MSC:
90C55Methods of successive quadratic programming type
49J20Optimal control problems with PDE (existence)
49J15Optimal control problems with ODE (existence)
49M37Methods of nonlinear programming type in calculus of variations
65F05Direct methods for linear systems and matrix inversion (numerical linear algebra)
65K05Mathematical programming (numerical methods)
90C30Nonlinear programming