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.

SNOPT is a general-purpose system for solving optimization problems involving many variables and constraints. It minimizes a linear or nonlinear function subject to bounds on the variables and sparse linear or nonlinear constraints. It is suitable for large-scale linear and quadratic programming and for linearly constrained optimization, as well as for general nonlinear programs. SNOPT finds solutions that are locally optimal , and ideally any nonlinear functions should be smooth and users should provide gradients. It is often more widely useful. For example, local optima are often global solutions, and discontinuities in the function gradients can often be tolerated if they are not too close to an optimum. Unknown gradients are estimated by finite differences. SNOPT uses a sequential quadratic programming (SQP) algorithm that obtains search directions from a sequence of quadratic programming subproblems. Each QP subproblem minimizes a quadratic model of a certain Lagrangian function subject to a linearization of the constraints. An augmented Lagrangian merit function is reduced along each search direction to ensure convergence from any starting point. SNOPT is most efficient if only some of the variables enter nonlinearly, or if the number of active constraints (including simple bounds) is nearly as large as the number of variables. SNOPT requires relatively few evaluations of the problem functions. Hence it is especially effective if the objective or constraint functions (and their gradients) are expensive to evaluate. The source code for SNOPT is suitable for any machine with a Fortran compiler. SNOPT may be called from a driver program (typically in Fortran, C or MATLAB). SNOPT can also be used as a stand-alone package, reading data in the MPS ...

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An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN ). The linear equality constraints are handled using an exact L 1 penalty function which is made smooth in the same way as the Euclidean norms. It is shown that the dual problem is to maximize a linear objective function subject to homogeneous linear equality constraints and quadratic inequalities. Hence the suggested method also solves such problems efficiently. In fact such a problem from plastic collapse analysis motivated this work. Numerical results are presented for large sparse problems, demonstrating the extreme efficiency of the method. Keywords: Sum of Norms, Nonsmooth Optimization, Duality, Newton Barrier Method. AMS(MOS) subject classification: 65K05, 90C06, 90C25, 90C90. Abbreviated title: A Newton barrier method. Supported by the ...

. 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...

Introduction This section describes the GAMS interface to the general-purpose NLP solver SNOPT, (Sparse Nonlinear Optimizer) which implements a sequential quadratic programming (SQP) method for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. The optimization problem is assumed to be stated in the form NP minimize x or maximize x f(x) subject to F (x) # b 1 Gx # b 2 l # x # u, (1) where x # # n , f(x) is a linear or nonlinear smooth objective function, l and u are constant lower and upper bounds, F (x) is a set of nonlinear constraint functions, G is a sparse ma

. Exact penalty methods for the solution of constrained optimization problems are based on the construction of a function whose unconstrained minimizing points are also solution of the constrained problem. In the first part of this paper we recall some definitions concerning exactness properties of penalty functions, of barrier functions, of augmented Lagrangian functions, and discuss under which assumptions on the constrained problem these properties can be ensured. In the second part of the paper we consider algorithmic aspects of exact penalty methods; in particular we show that, by making use of continuously differentiable functions that possess exactness properties, it is possible to define implementable algorithms that are globally convergent with superlinear convergence rate towards KKT points of the constrained problem. 1 Introduction "It would be a major theoretic breakthrough in nonlinear programming if a simple continuously differentiable function could be exhibited with th...

We propose a quasi-Newton algorithm for solving large optimization problems with nonlinear equality constraints. It is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations. The algorithm incorporates a correction vector that approximates the cross term Z T I/VYp in order to estimate the curvature in both the range and null spaces of the constraints. The algorithm can be considered to be, in some sense, a practical implementation of an algorithm of Coleman and Conn. We give conditions under which local and superlinear convergence is obtained.

Using solver specific options 1