Search

In this paper, a new line search filter algorithm for equality constrained optimization is presented. The approach belongs to the class of inexact Newton-like methods. It can also be regarded as an inexact version of generic sequential quadratic programming (SQP) methods. The trial step is obtained by truncatedly solving the primal-dual system based on any robust and efficient linear system solver. Practical termination tests for the linear system solver are established to ensure global convergence. Preliminary numerical results demonstrate the approach is potentially useful.

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy and efficiency of our algorithm.

In this work adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner. The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction-diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice.

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.

Two trust-region methods for systems of mixed nonlinear equalities, general inequalities and simple bounds are proposed. The first method is based on a Gauss–Newton model, the second one is based on a regularized Gauss–Newton model and results to be a Levenberg–Marquardt method. The globalization strategy uses affine scaling matrices arising in bound-constrained optimization. Global convergence results are established and quadratic rate is achieved under an error bound assumption. The numerical efficiency of the new methods is experimentally studied.

In this paper, by introducing a new nonmonotone line search technique and combining it with the spectral projected gradient method, we propose a solution method for solving the bound constrained optimization problems. Under mild conditions, we obtain the global convergence even without requiring a prior of the existence of a limit point. Numerical tests are also given to show the efficiency of the proposed method.

The problems of optimization of cylindrical bar cross-sections are formulated in variational forms. The functional considered characterizes torsional and bending rigidities, and the area of cross-section of the bar. The shape of the boundary of the cross-section is taken as a design variable. The problem is first expressed as an optimal control problem. Then by using an embedding method, the class of admissible shapes is replaced by a class of positive Borel measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this finite dimensional linear programming problem. Numerical examples are also given.

We consider the problem of planar grid generation. This problem can be easily reformulated as an unconstrained optimization problem with the usual variational method. This is a very simple technique to produce grids on planar domains, but it cannot be used in automatic grid generation codes. As a matter of fact, its formulation depends on some parameters, whose right value can disagree considerably among different domains. A simple modification of the variational method is proposed. This modification is mainly based on a linearly constrained optimization problem and it gives a significant increase of the robustness of the method.

In recent papers, the problem of estimating the thickness and the optical constants (refractive index and absorption coefficient) of thin films using only transmittance data has been addressed by means of optimization techniques. Models were proposed for solving this problem using linearly constrained optimization and unconstrained optimization. However, the optical parameters of "very thin" films could not be recovered with methods that are successful in other situations. Here we introduce an optimization technique that seems to be efficient for recovering the parameters of very thin films.

This paper presents an iterative method for solving bound-constrained systems of nonlinear equations. It combines ideas from the classical trust-region Newton method for unconstrained nonlinear equations and the recent interior affine scaling approach for constrained optimization problems. The method generates feasible iterates and handles the bounds implicitly. It reduces to a standard trust-region method for unconstrained problems when there are no upper or lower bounds on the variables. Global and local fast convergence properties are obtained. The numerical performance of the method is shown on a large number of test problems.

A reconstruction technique for estimating boundary conditions in natural waters from in situ radiance data is presented. The inverse problem is formulated as a nonlinear constrained optimization problem. The objective function is defined as the square Euclidean norm of the difference vector between experimental and computed data. The associated direct problem is solved by LTSN method.

In this paper an algorithm for identifying an index 1 or 2 differential–algebraic subsystem from a possibly higher index path-constrained dynamical system is proposed. The algorithm is useful for diagnostic purposes in model development, and in the formulation of dynamic optimization problems to be solved by shooting or multiple shooting type methods.

Constrained optimization problems occur in many applications of engineering, science and medicine. Much attention has recently been devoted to solving this class of problems using trust region algorithms with strong convergence properties, in part because of the availability of reliable software. This paper presents a survey of recent advances in trust region algorithms. We then explain the different choices of penalty function, Lagrange function and expanded Lagrangian function used for modeling constrained optimization problems and solving these equations using trust region algorithms. Finally, some numerical results for the implementation of our proposed method on different test problems with various sizes are presented.

A modified Newton-like method for nonlinearly constrained optimization calculations is presented and applied to a variety of separation process optimization problems. The nonlinear programming algorithm in this paper is similar to those of Wilson, Han and Powell. It differs only in the way in which approximations to the Hessian matrix of the Lagrangian function are built. For separation process applications, a large amount of reliable second-derivative information is available in analytical form. Much of the remaining second-derivative information is subject to certain thermodynamic constraints. Our algorithm exploits these two facts by building Hessian matrix approximations in two parts, a computed part which is built analytically, and an approximated part which is built by iterated projections and the Powell-symmetric-Broyden formula. Iterated projections are required so that the quasi-Newton matrix approximations satisfy the thermodynamic constraints at each iteration and a limiting secant condition. Further advantage is taken of the complete separability of the thermodynamic functions. This modified successive quadratic programming algorithm has been compared to various Newton-like methods on a number of separation process optimization problems. These include cost minimization calculations for one- and two-stage flash units and the optimization of the operating conditions of a distillation column. The numerical results show that our modified algorithm can compete favorably with existing methods, in terms of reliability and efficiency, on these applications.