Three new algorithms based on the sequential secant method

We introduce three new implementations of the sequential secant method for solving nonlinear simultaneous equations. Following the ideas of Gragg and Stewart, we store orthogonal factorizations of some of the matrices involved. Degeneracy in the increments of the independent variable is corrected according to simple and theoretically justified procedures. Some numerical experiences are also given.     

A direct method of linearization for continuous minimax problems

We consider the problem of minimizing a nondifferentiable function that is the pointwise maximum over a compact family of continuously differentiable functions. We suppose that a certain convex approximation to the objective function can be evaluated. An iterative method is given which uses as successive search directions approximate solutions of semi-infinite quadratic programming problems calculated via a new generalized proximity algorithm. Inexact line searches ensure global convergence of the method to stationary points.This work was supported by Project No. CPBP-02.15/2.1.1.     

An active set strategy for solving optimization problems with up to 200,000,000 nonlinear constraints

Numerical test results are presented for solving smooth nonlinear programming problems with a large number of constraints, but a moderate number of variables. The active set method proceeds from a given bound for the maximum number of expected active constraints at an optimal solution, which must be less than the total number of constraints. A quadratic programming subproblem is generated with a reduced number of linear constraints from the so-called working set, which is internally changed from one iterate to the next. Only for active constraints, i.e., a certain subset of the working set, new gradient values must be computed. The line search is adapted to avoid too many active constraints which do not fit into the working set. The active set strategy is an extension of an algorithm described earlier by the author together with a rigorous convergence proof. Numerical results for some simple academic test problems show that nonlinear programs with up to 200,000,000 nonlinear constraints are efficiently solved on a standard PC.     

A heuristic algorithm for nonlinear programming

In this paper, a heuristic algorithm for nonlinear programming is presented. The algorithm uses two search directions, and the Hessian of the Lagrangian function is approximated with the BFGS secant update. We show that the sequence of iterates convergeq-superlinearly if the sequence of approximating matrices satisfies a particular condition. Numerical results are presented.     

A nonlinear interval number programming method for uncertain optimization problems

In this paper, a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, in which the robustness of design is considered. Through a modified possibility degree, the uncertain inequality and equality constraints are changed to deterministic inequality constraints. The two objective functions are converted into a single-objective problem through the linear combination method, and the deterministic inequality constraints are treated with the penalty function method. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimization problem. Two numerical examples are investigated to demonstrate the effectiveness of the present method.     

解变分不等式问题的混合方法

By using Fukushima‘s differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993. In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty, polyhedral,closed and convex is proposed. Armijo-type line search and trust region strategies as well as Fukushima‘s differentiable merit function are incorporated into the method. It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al. ,the method reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experiences show the efficiency of the proposed method.     

A Structured Secant Method Based on a New Quasi-Newton Equation for Nonlinear Least Squares Problems

In this paper, a new quasi-Newton equation is applied to the structured secant methods for nonlinear least squares problems. We show that the new equation is better than the original quasi-Newton equation as it provides a more accurate approximation to the second order information. Furthermore, combining the new quasi-Newton equation with a product structure, a new algorithm is established. It is shown that the resulting algorithm is quadratically convergent for the zero-residual case and superlinearly convergent for the nonzero-residual case. In order to compare the new algorithm with some related methods, our preliminary numerical experiments are also reported.     

Generalized Newton method for linear optimization problems with inequality constraints

A dual problem of linear programming is reduced to the unconstrained maximization of a concave piecewise quadratic function for sufficiently large values of a certain parameter. An estimate is given for the threshold value of the parameter starting from which the projection of a given point to the set of solutions of the dual linear programming problem in dual and auxiliary variables is easily found by means of a single solution of the unconstrained maximization problem. The unconstrained maximization is carried out by the generalized Newton method, which is globally convergent in an a finite number of steps. The results of numerical experiments are presented for randomly generated large-scale linear programming problems.     

A unified framework for linear control problems with state-variable inequality constraints

This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state-variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.This research was partially supported by Grant No. A4619 from the National Research Council of Canada to the first author. The first author also acknowledges the support provided by the Brookhaven National Laboratory, where he conducted his research.     

A robust algorithm for quadratic optimization under quadratic constraints

Most existing methods of quadratically constrained quadratic optimization actually solve a refined linear or convex relaxation of the original problem. It turned out, however, that such an approach may sometimes provide an infeasible solution which cannot be accepted as an approximate optimal solution in any reasonable sense. To overcome these limitations a new approach is proposed that guarantees a more appropriate approximate optimal solution which is also stable under small perturbations of the constraints.     

A robust sequential quadratic programming method

The sequential quadratic programming method developed by Wilson, Han and Powell may fail if the quadratic programming subproblems become infeasible, or if the associated sequence of search directions is unbounded. This paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems. Furthermore, questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed.Work supported in part by the National Science Foundation under Grant No. DMS-8602399 and by the Air Force Office of Scientific Research under Grant No. ISSA-860080.Work supported in part by the National Science Foundation under Grant No. DMS-8602419.     

A family of quasi-Newton methods for unconstrained optimization problems

ABSTRACT

In this paper, we present a class of approximating matrices as a function of a scalar parameter that includes the Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno methods as special cases. A powerful iterative descent method for finding a local minimum of a function of several variables is described. The new method maintains the positive definiteness of the approximating matrices. For a region in which the function depends quadratically on the variables, no more than n iterations are required, where n is the number of variables. A set of computational results that verifies the superiority of the new method are presented.     

Surrogate duality for robust optimization

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min–max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, and show some examples at which such duality results are used effectively. Moreover, we obtain a surrogate duality theorem and a surrogate min–max duality theorem for semi-definite optimization problems in the face of data uncertainty.     

A secant algorithm with line search filter method for nonlinear optimization

Filter methods were initially designed for nonlinear programming problems by Fletcher and Leyffer. In this paper we propose a secant algorithm with line search filter method for nonlinear equality constrained optimization. The algorithm yields the global convergence under some reasonable conditions. By using the Lagrangian function value in the filter we establish that the proposed algorithm can overcome the Maratos effect without using second order correction step, so that fast local superlinear convergence to second order sufficient local solution is achieved. The primary numerical results are presented to confirm the robustness and efficiency of our approach.     

A new technique for inconsistent QP problems in the SQP method

Successful treatment of inconsistent QP problems is of major importance in the SQP method, since such occur quite often even for well behaved nonlinear programming problems. This paper presents a new technique for regularizing inconsistent QP problems, which compromises in its properties between the simple technique of Pantoja and Mayne [36] and the highly successful, but expensive one of Tone [47]. Global convergence of a corresponding algorithm is shown under reasonable weak conditions. Numerical results are reported which show that this technique, combined with a special method for the case of regular subproblems, is quite competitive to highly appreciated established ones.     

Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization

Augmented Lagrangian methods for large-scale optimization usually require efficient algorithms for minimization with box constraints. On the other hand, active-set box-constraint methods employ unconstrained optimization algorithms for minimization inside the faces of the box. Several approaches may be employed for computing internal search directions in the large-scale case. In this paper a minimal-memory quasi-Newton approach with secant preconditioners is proposed, taking into account the structure of Augmented Lagrangians that come from the popular Powell–Hestenes–Rockafellar scheme. A combined algorithm, that uses the quasi-Newton formula or a truncated-Newton procedure, depending on the presence of active constraints in the penalty-Lagrangian function, is also suggested. Numerical experiments using the Cute collection are presented.     

Equality and inequality constrained optimization algorithms with convergent stepsizes

In this paper, we consider two algorithms for nonlinear equality and inequality constrained optimization. Both algorithms utilize stepsize strategies based on differentiable penalty functions and quadratic programming subproblems. The essential difference between the algorithms is in the stepsize strategies used. The objective function in the quadratic subproblem includes a linear term that is dependent on the penalty functions. The quadratic objective function utilizes an approximate Hessian of the Lagrangian augmented by the penalty functions. In this approximation, it is possible to ignore the second-derivative terms arising from the constraints in the penalty functions.The penalty parameter is determined using a strategy, slightly different for each algorithm, that ensures boundedness as well as a descent property. In particular, the boundedness follows as the strategy is always satisfied for finite values of the parameter.These properties are utilized to establish global convergence and the condition under which unit stepsizes are achieved. There is also a compatibility between the quadratic objective function and the stepsize strategy to ensure the consistency of the properties for unit steps and subsequent convergence rates.This research was funded by SERC and ESRC research contracts. The author is grateful to Professors Laurence Dixon and David Mayne for their comments. The numerical results in the paper were obtained using a program written by Mr. Robin Becker.     

A new local and global optimization method for mixed integer quadratic programming problems

In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable.