Abstract: This paper is concerned with the Stackelberg problem and the min-max problem in competitive systems. The Stackelberg approach is applied to the optimization of two-level systems where the higher level determines the optimal value of its decision variables (parameters for the lower level) so as to minimize its objective, while the lower level minimizes its own objective with respect to the lower level decision variables under the given parameters. Meanwhile, the min-max problem is to determine a min-max solution such that a function maximized with respect to the maximizer's variables is minimized with respect to the minimizer's variables. This problem is also characterized by a parametric approach in a two-level scheme. New computational methods are proposed here; that is, a series of nonlinear programming problems approximating the original two-level problem by application of a penalty method to a constrained parametric problem in the lower level are solved iteratively. It is proved that a sequence of approximated solutions converges to the correct Stackelberg solution, or the min-max solution. Some numerical examples are presented to illustrate the algorithms.

Abstract: A new method for solving discrete structural optimization problems is presented. An interior penalty function is used to convert the original constrained problem into an unconstrained parametric problem. Then the search for the optimal solution to the parametric problem is based on a discrete direction gradient. Solving an appropriate sequence of these unconstrained parametric problems is equivalent to solving the original constrained optimization problem. This method is illustrated first on a small reinforced concrete problem, and then to the design of steel building frames which are made up of standard sections. Results for a one-story four-bay unsymmetrical frame and an eight-story three-bay symmetrical frame are described.

Abstract: Recently developed methods for nonlinear semi-infinite programming problems have only local convergence properties. In this paper, we show how the convergence can be globalized by the use of an exact penalty function. Both convergence and rate of convergence results are established.

Abstract: There is currently much interest in solving nonlinear programming problems by SOLVER-like methods in which a quadratic programming (QP) program is solved on each iteration. When used in conjunction with a line search good numerical evidence is often reported. However this paper points out that these methods can fail and an example is given. A new algorithm is investigated which minimizes the associated exact L1 penalty function. By making certain linear and quadratic approximations a QP-like subproblem is determined which is not significantly more complicated than the standard QP problem. When used in conjunction with a trust region strategy the resulting algorithm is globally convergent with no unrealistic assumptions. Usually the algorithm is equivalent to the SOLVER method close to the solution so the advantages of the latter method are retained, including the second order rate of convergence. A second algorithm is also investigated which estimates the active constraint set and so avoids the QP-like subproblem, and which can also be implemented with only n2 + 0(n) storage. Numerical evidence with both algorithms is reported. The first algorithm appears to be comparable with the SOLVER method but is more robust in that solutions to some difficult problems are obtained. The second algorithm is less good for inequality constraints but has promise for solving equation problems.

Abstract: A class of generalized variable penalty formulations for solving nonlinear programming problems is presented The method poses a sequence of unconstrained optimization problems with mechanisms to control the quality of the approximation for the Hessian matrix, which is expressed in terms of the constraint functions and their first derivatives The unconstrained problems are solved using a modified Newton's algorithm The method is particularly applicable to solution techniques where an approximate analysis step has to be used (eg, constraint approximations, etc), which often results in the violation of the constraints The generalized penalty formulation contains two floating parameters, which are used to meet the penalty requirements and to control the errors in the approximation of the Hessian matrix A third parameter is used to vary the class of standard barrier or quasibarrier functions, forming a branch of the variable penalty formulation Several possibilities for choosing such floating parameters are discussed The numerical effectiveness of this algorithm is demonstrated on a relatively large set of test examples

Abstract: : This paper presents an algorithm for the minimization of a nonlinear objective function subject to nonlinear inequality and equality constraints. The proposed method has the two distinguishing properties that, under weak assumptions, it converges to a Kuhn-Tucker point for the problem and under somewhat stronger assumptions, the rate of convergence is quadratic. The method is similar to a recent method proposed by Rosen in that it begins by using a penalty function approach to generate a point in a nighborhood of the optimum and then switches to Robinson's method. The new method has two new features not shared by Rosen's method. First, a correct choice of penalty function parameters is constructed automatically, thus guaranteeing global convergence to a stationary point. Second, the linearly constrained subproblems solved by the Robinson method normally contain linear inequality constraints while for the method presented here, only linear equality constraints are required. That is, in a certain sense, the new method 'knows' which of the linear inequality constraints will be active in the subproblems. The subproblems may thus be solved in an especially efficient manner. Preliminary computational results are presented. (Author)

Abstract: We consider solving a minimization convex program by sequentially solving a minimization convex approximating subproblem and then executing a line search on an exact penalty function. Each subproblem is constructed from the current estimate of a solution of the given problem, possibly together with other information. Under mild conditions, solving the current subproblem generates a descent direction for the exact penalty function. Minimizing the exact penalty function along the current descent direction provides a new estimate of a solution, and a new subproblem is formed. For any arbitrary starting estimate, this scheme generates a sequence of estimates that converges to a solution of the given problem. Moreover, the functions defining the given problem and each subproblem need not be differentiable.

Abstract: The dimensional synthesis of a four bar linkage for function generation, usually performed by considering only kinematic characteristics of the mechanism, may be improved by accounting also for its dynamical behavior In the present study the maximum values of velocity and acceleration are controlled in order to limit the inertial forces, as required in the design of high-speed mechanisms At this purpose an optimization technique, based on penalty function method, has been adopted Several numerical results for different choices for constraints and initial design are presented and discussed

Abstract: Based on the multiplier method of constrained minimization, an algorithm is developed to handle the constrained estimation problem in covariance structure analysis. In the context of a general model which has wide applicability in multivariate medical and behavioural researches, computer programs are implemented to produce the weighted least squares estimates and the maximum likelihood estimates. The multiplier method is compared with the penalty function method in terms of computer time, number of iterations and number of unconstrained minimizations. The indication is that the multiplier method is substantially better.

Abstract: We use the finite element method to compute optimal controls of systems governed by linear ordinary differential equations with a quadratic performance index. As an application we use the penalty technique to solve terminal state optimal controllability problems. Numerical instabilities, which are common in the use of penalty, are minimized when the finite element method is applied to solve this problem. Convergence theorems are given and error and penalty parameter estimates are presented. Concrete examples for various situations are given to illustrate the theory.

Abstract: The existence of an initial non-physical transient when using the penalty method in modeling the time-dependence incompressible Navier-Stokes equations is investigated theoretically and demonstrated numerically using the Galerkin finite element technique A stable, variable step time integration scheme which can overlook the initial non-physical transient while using reasonable-sized time steps is described Numerical examples illustrating the time integration scheme and concomitantly the difference in transient response of an incompressible fluid and its slightly compressible (penalty) analog are presented

Abstract: This study establishes an error estimate for a penalty-finite element approximation of the variational inequality obtained by a class of obstacle problems. By special identification of the penalty term, we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality. The rate of convergence of the penalization is ? where ? is the penalty parameter. To obtain the error of finite element approximation, we apply the results obtained by Brezzi, Hager and Raviart for the mixed finite element method to the variational inequality.

Abstract: This paper presents a general theory of exterior penalty methods for constrained optimization problems with applications to the analysis and approximation of certain problems in linear and nonlinear elasticity. Particular attention is given to the role of a “generalized LBB-condition” which governs the behavior of penalty approximations of Lagrange multipliers.

Abstract: Cracked plates of power hardening material under plane strain and incompressibility conditions are investigated in this work by using a penalty function and superposition method. The present numerical method is found to be quite efficient for the relevant problems if the hardening exponent of material is not so large.

Abstract: : The problem of shape optimization to minimize stress concentration factors is presented. Design variables are chosen such that optimal configurations can be produced for a cylindrical pressure vessel with an end closure and nozzle intersection. A system model is formed in which the stress gradients are calculated numerically using the finite elements method and optimization is performed by the penalty function technique. Various examples of the design algorithm are presented showing optimal geometries together with plots of boundary stress concentrations for the sequence of designs generated. (Author)

Abstract: It is shown that a norm penalty method is exact for mixed integer programs in rational data, in the sense that the minimization of the criterion plus penalty over the nonnegativities and integrality constraints has the same set of globally optimal solutions as does the mixed integer program with the equality constraints present. This result is then extended to mixed-integer programs with complementarity constraints. An example shows that no differentiable penalty can be exact for mixed integer programs.

Abstract: The paper deals with nonconforming finite element methods for the approximate solution of the interior boundary value problem for Maxwell equations in the time-harmonic case. The methods are based on penalization in the boundary conditions of total reflexion. Qualitative convergence results are obtained by a-priori estimates which are proven in the first part of this paper. The main object is to establish estimates for the global discretization error in various norms of the underlying spaces of approximating vector fields.

Abstract: One can transform an optimal reliability design problem for complex systems into an equivalent problem for which solutions can be derived in a much simpler and more efficient manner. This results in a resource allocation problem that can be solved by efficient techniques. The major assumption is that the original problem has only one associated constraining equation. However, this approach can be extended to problems with multiple constraints by applying the procedure in an iterative fashion within a penalty function framework. This approach applies to any system for which the reliability can be calculated. Each logical stage of the system under study can incorporate redundancy via a parallel arrangement, standby, or k-out-of-n:G.

Abstract: Using an appropriate active set strategy Han's method for solving a general differentiable nonlinear programming problem can be modified such that only equality constrained quadratic subproblems (i.e. linear equations) have to be solved in order to find a descent direction of the Zangwill-Pietrzykowski penalty function of the problem. The present paper gives a complete description of a correponding algorithm along with proofs of convergence and rate of convergence results. A crucial assumption concerns the linear independence of the gradients of active constraints which makes return to the feasible set possible whenever desirable. Under this assumption every cluster point of the sequence generated is a Kuhn-Tucker point. If the sequence comes near to a Kuhn-Tucker point where second order sufficiency holds with strict complementary slackness, convergence is Q-superlinear in the primal and the dual variables, if one uses consistent approximations of the Hessian of the Lagrangian. If second order derivatives are used, convergence is second order of course.

Abstract: Robinson's quadratically convergent algorithm for general nonlinear programming problems is modified in such a way that, instead of exact derivatives of the objective function and the constraints, approximations of these can be used which are computed by differences of function values. This locally convergent algorithm is then combined with a penalty function method to provide a globally and quadratically convergent algorithm that does not require the calculation of derivatives.