Abstract: SUMMARY The frequently used reduced integration method for solving incompressible flow problems 'a la penalty' is critically examined vis-a-vis the consistent penalty method. For the limited number of quadrilateral and hexahedral elements studied, it is shown that the former method is only equivalent to the latter in certain special cases. In the general case, the consistent penalty method is shown to be more accurate. Finally, we demonstrate significant advantages of a new element, employing biquadratic (2-D) or triquadratic (3-D) velocity and linear pressure over that using the same velocity but employing bilinear (2-D) or trilinear (3-D) pressure approximation.

Abstract: The purpose of this paper is to derive, in a unified way, second order necessary and sufficient optimality criteria, for four types of nonsmooth minimization problems: thediscrete minimax problem, thediscrete l1-approximation, the minimization of theexact penalty function and the minimization of theclassical exterior penalty function. Our results correct and supplement conditions obtained by various authors in recent papers.

Abstract: We present a globally and superlinearly converging method for equality-constrained optimization requiring the updating of a reduced size matrix approximating a restriction of the Hessian of the Lagrangian. Each iterate is obtained by a search along a simple curve defined by a quasi-Newton direction and a feasibility improving direction; an exact penalty function is used to determine the stepsize. The method can be viewed as an efficient approximation to the quasi-Newton along geodesics of [1] where feasibility was enforced at each step. Its relation with multiplier methods and recursive quadratic programming methods is also investigated.

Abstract: In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.

Abstract: The paper presents a smoothing technique to the oscillated contact pressure obtained by penalty methods for a class of unilateral contact problems in linear elasticity. The main result is to show that the smoothed contact pressure satisfies the so-called Babuska–Brezzi condition which dominates the convergence of the penalty method. One numerical example is described.

Abstract: : This document is the final technical report for the project, Computational Methods in Nonlinear Mechanics. The Report summarizes results obtained on this project for the period June 30, 1980 through September 30, 1982. All objectives of the original statement of work have been accomplished. These have included advances in mathematical modelling, numerical analysis, approximation theory, development of computer codes, and the study of related issues connected with the following nonlinear structural problems: contact problems in elasticity, reduced integration and penalty methods for finite element approximations of constrained problems in elasticity and fluid flow, contact problems with friction, nonlinear friction laws, plasticity and metal forming, existence and approximation theories in contact problem in elastostatics, fracture mechanics numerical methods for problems of finite elastic deformation, nonlinear eigenvalue problems, and bifurcation theory. Detailed summaries of results in some of these areas are given together with lists of all papers, reports, books, dissertations and oral presentations produced during the contract period. Suggestions for further research areas are also given. (Author)

Abstract: The equivalence of zero–one integer programming and a concave quadratic penalty function problem has been shown by Raghavachari, for a sufficiently large value of the penalty. A lower bound for this penalty is obtained here, which in specific cases cannot be reduced.

Abstract: The use of interior penalty methods as a basis for developing finite element approximations of boundary value problems with constraints is explored. Particular attention is given to the Signorini problem of contact of an elastic body with a rigid foundation. Error estimates are derived and the results of a numerical experiment are discussed.

Abstract: In this paper, we consider Newton's method for solving the system of necessary optimality conditions of optimization problems with equality and inequality constraints. The principal drawbacks of the method are the need for a good starting point, the inability to distinguish between local maxima and local minima, and, when inequality constraints are present, the necessity to solve a quadratic programming problem at each iteration. We show that all these drawbacks can be overcome to a great extent without sacrificing the superlinear convergence rate by making use of exact differentiable penalty functions introduced by Di Pillo and Grippo (Ref. 1). We also show that there is a close relationship between the class of penalty functions of Di Pillo and Grippo and the class of Fletcher (Ref. 2), and that the region of convergence of a variation of Newton's method can be enlarged by making use of one of Fletcher's penalty functions.

Abstract: In this paper, we consider the nonlinear programming problem P to minimize f(x) subject to g i (x)<=0 for i=1, …, m and x∈X. If X is compact and the number of global optimal solutions is finite, under a suitable constraint qualification, we show that a globally exact penalty function exists. Particularly, we establish a one to one correspondence between global optimal solutions to the original problem and global minimizers of the penalty problem for a sufficiently large, but finite, penalty parameter. A lower bound on the penalty parameter is provided in terms of the Kuhn-Tucker Lagrangian multipliers and lower bounds on the functions involved.

Abstract: The recently proposed quasi-Newton method for constrained optimization has very attractive local convergence properties. To force global convergnce of the method, a descent method which uses Zangwill's penalty function and an exact line search has been proposed by Han. In this paper a new method which adopts a differentiable penalty function and an approximate line is presented. The proposed penalty function has the form of the augmented Lagrangian function. An algorithm for updating parameters which appear in the penalty function is described. Global convergence of the given method is proved.

Abstract: 1: Mathematical Programming and Optimal Control Theory.- An Optimality Condition and its Application to Parametric Semi-Infinite Optimization.- The Choice of a Parameter in a Penalty Method.- Recent Results on ?-Conjugation and Nonconvex Optimization.- On Quantitative Stability of Point-to-Set-Mappings and the Rate of Convergence of Corresponding Algorithms.- On the Penalization Method in Convex Stochastic Programming.- A New Algorithm of Solving the Flow - Shop Problem.- On Dynamic Traffic Assignment.- On an Approximation Problem of Mechanical Structural Optimization.- Optimal Daily Scheduling of the Electricity Production in Hungary.- Power Distribution Planning and the Application of Linear Mixed-Integer Programming.- Optimal Flood Control by Reservoir Systems Using the Reduced Gradient Method.- Instant Optimization of Hydro Energy Storage Plants.- Dynamic Programming in Power System Extension Planning.- Some New Multicriteria Approaches.- Equilibrium Selection in a Wage Bargaining Situation with Incomplete Information.- Planning and Forecast Horizons in a Simple Wheat Trading Model.- Intertemporal Reversales of Environmental and Macroeconomic Policies.- Optimal Control of Concave Economic Models with two Control Instruments.- Optimal Control with Switching Dynamics.- Dynamic Systems with Several Decision-Makers.- Optimal Bimodal Harvest Policies in Age-Specific Bioeconomic Models.- Growth Rates, Optimal Harvesting and Related Topics in the Mass Rearing of Tsetse Flies.- The Release of Partly Fertile Males or Females in the Application of the Sterile-Insect Technique: Mathematical Analysis of the Hard-Release Strategy.- 2: Stochastic Models.- New Developments in Optimal Control of Queueing Systems.- Estimation and Control in a GI|M|1-System.- On Discriminating among Stochastic Models - A Survey.- Increasing the Work-Safety in Nuclear Power Plants through the Use of Preventive Maintenance Policies.- Recent Developments in Econometrics.- Slight Misspecifications of Linear Systems.- Local Sensitivity Analysis and Matrix Derivatives.- Analysis and Forecasting of Demand for Electricity Using Time Series Analysis.- Short Term Load Predication in Electric Power Systems.- Interactive Short-Term Load Forecasting.- Predicting the Demand for Electricity - An Application of Transfer Function Analysis.- Problems Associated with the Design of a Reliability Model in Electricity Industry.

Abstract: The traditional method of data fitting is by the least squares (l2) technique. When the data is good—reasonably accurate with normally distributed errors—this method is ideal. When the data is bad—contaminated by occasional wild values—then the l1 technique (minimizing sums of absolute values of residuals) has much to recommend it. This paper surveys the strategy of a globally and superlinearly convergent algorithm to minimize sums of absolute values of C2 functions. The approach to be presented is closely related to the use of a certain, piecewise differentiable penalty function to solve nonlinear programming problems.

Abstract: Several algorithms for constrained optimization are based on the idea of choosing the search direction as the solution of a quadratic program (QP). However, there is considerable variation in the precise nature of the quadratic program to be solved. Furthermore, significant differences exist in the procedures advocated to ensure that the search direction is well defined, and some algorithms abandom the quadratic programming approach for particular iterations under certain conditions. In this paper, we discuss some crucial differences in formulation and solution that arise in QP-based methods for linearly constrained and nonlinearly constrained optimization, with particular emphasis on the treatment of inequality constraints. For linearly constrained problems, we consider the effect of formulating the constraints of the QP sub-problem as equalities or inequalities. In the case of nonlinear constraints, the issues to be discussed include incompatibility or ill-conditioning of the constraints, determination of the active set, Lagrange multiplier estimates, and approximation of the Lagrangian function.

Abstract: This chapter discusses the method of multipliers for equality constrained problems. By solving an approximate problem, an approximate solution of the original problem can be obtained. However, if a sequence of approximate problems can be constructed that converges in a well-defined sense to the original problem, then the corresponding sequence of approximate solutions would yield in the limit a solution of the original problem. The basic idea in penalty methods is to eliminate some or all of the constraints and add to the objective function a penalty term that prescribes a high cost to infeasible points. A parameter that determines the severity of the penalty and as a consequence the extent to which the resulting unconstrained problem approximates the original constrained problem is associated with the penalty methods.

Abstract: A technique is described for solving generalized geometric programs whose constraints include one or more strict equalities. The algorithm solves a sequence of penalized geometric programs; the penalty functions are derived from the arithmetic-geometric inequality as condensed posynomials. Two examples serve to illustrate the idea.

Abstract: A comprehensive comparative study of nonlinear programming algorithms as applied to engineering design is presented. Linear approximation methods, interior penalty function methods and exterior penalty function methods were tested on a set of thirty problems and were rated on their ability to solve problems within a reasonable amount of computational time. The effect of the problem parameters on the solution time for the various classifications of algorithms was studied. The variable parameters included the number of design variables, the number of inequality constraints, the number of equality constraints and the degree of nonlinearity of the objective function and constraints. Also a combined penalty function and linear approximation algorithm was investigated.

Abstract: A mathematical analysis of the penalty method applied to the initial boundary value problem for the Stokes equations is presented. We prove a regularity result for the penalty equations by means of a spectral Galerkin method. Based upon this result the discretization in time (by a multistep integration formula) and in space (by a finite element method) is shown to give an optimal L 2-error estimate.

Abstract: Penalty function methods have been widely used for solving constrained nonlinear programming problems that arise in optimal load-flow analysis of power systems. One of the basic drawbacks of these methods is that the Hessian of the penalty function becames ill-conditioned as-the penalty factor takes very large (or small) values. Conseqpently minimisation of the penalty function becomes rather a difficult task even for an efficient optimiser.

Abstract: Let Ω be a bounded open subset of R n; with every measurable subset A of Ω is associated the real number En(A), related to capacity (for electrostatic problems) or to heat flux (for problems of thermal conduction). One investigates the minimization of En(A) when the measure of A is imposed. Penalty methods allows one to apply some techniques of convex analysis.

Abstract: On a model of clamped beam problem we study a finite element method with approximate constraints and the corresponding penalty function approach. By a small perturbation analysis we show that the rotation is independent from the penalization. This method allows us to use simple c0 elements, hence we can compute approximate solutions to obstacle problems. The one dimensional nature of all the results is underlined and the extension of this FEM to flexible pipe lines is indicated.

Abstract: The quadratic penalty function is widely used in the practical implementations of methods of multipliers. There is a tangible advantage in using a different penalty function. The objective function is bounded below along the constraint set and the augmented Lagrangian is unbounded over the entire space for every value of the penalty parameter. The augmented Lagrangian functions for inequality constraints and some of the approximating functions do not have continuous second derivatives. The methods to be used for unconstrained minimization of the augmented Lagrangian rely on the continuity of second derivatives. Multiplier methods corresponding to different types of penalty functions can exhibit different rates of convergence.