Optimal controls with equality state constraints

In this paper, we derive necessary conditions for a measurable controlu with values in a compact convex subsetU ofR ^P to minimize the integral $$\int_a^b {b^0 (s,x(s),u(s))} ds$$ subject to equality and inequality state constraints. The general conditions take the form of a maximum principle which involves an integration with respect to a finitely additive set function, but conditions are given under which this reduces to the classical maximum principle. These results are similar to results derived in Ref. 1 for relaxed controls.     

ODE methods for optimization with equality constraints

In an optimization problem with equality constraints we define an accessory function that is similar but different from a normal penalty function. In the accessory function we demonstrate the need to use small values of the parameter associated with an equality constraint. Large values of the parameter create extraneous stationary points which destroy the global convergence properties of steepest descent methods. By using small values of the parameters in the accessory function, when the current point is far away from the solution and when the constraint violations are large we are led to a refined version of the established SUMT method.     

Handling equality constraints with agent-based memetic algorithms

In addition to inequality constraints, many mathematical models require equality constraints to represent the practical problems appropriately. The existence of equality constraints reduces the size of the feasible space significantly, which makes it difficult to locate feasible and optimal solutions. This paper presents a new equality constraint handling technique which enhances the performance of an agent-based evolutionary algorithm in solving constrained optimization problems with equality constraints. The technique is basically used as an agent learning process in the agent-based evolutionary algorithm. The performance of the proposed algorithm is tested on a set of well-known benchmark problems including seven new problems. The experimental results confirm the improved performance of the proposed technique.     

Optimum design of wing structures with multiple frequency constraints

In this paper, design optimization of aircraft wing structures with multiple frequency constraints was considered. An optimality criterion algorithm along with a scaling procedure was used. Large-scale structural design problems were considered for demonstrating the reliability and efficiency of the algorithm. A simplified fighter wing, and an intermediate-complex wing were considered as design examples. Design histories and the first few frequencies at the initial and final conditions are presented.     

Least squares with non-linear equality constraints Application to closing of balances

It is shown how a least squares problem subject to equality constraints can be replaced by an unconstrained least squares problem. Constraints and equations may be non linear. Results seem to be too complicated to be applied to general cases but can quite successfully be used for special problems like the closing of balances for instance.     

Minimum-weight design of continuous beams under displacement and stress constraints

Explicit optimality conditions for minimum-weight design of elastic sandwich beams with segmentwise constant structural stiffness, subject to displacement and mean-square stress constraints, are obtained. An iterative procedure that combines the use of the optimality conditions with finite-element analysis is proposed and is illustrated by numerical examples. These examples suggest that very few iterations are necessary to obtain a good approximation to the optimal design. It is shown that, for practical purposes, the optimization problem may be simplified by using the optimality conditions derived for statically determinate beams instead of those valid for statically indeterminate beams.This research was sponsored by the U.S. Army Research Office, Durham, North Carolina. The authors are grateful to Professor W. Prager for helpful comments.     

Bayesian comparison of models with inequality and equality constraints

A Bayesian model selection procedure for comparing models subject to inequality and/or equality constraints is proposed. An encompassing prior approach is used, and a general form of the Bayes factor of a constrained model against the encompassing model is derived. A simple estimation method is proposed which can estimate the Bayes factors for all candidate models simultaneously by using one set of samples from the encompassing model. A simulation study and a real data analysis demonstrate performance of the method.     

Second-order conditions for extremum problems with nonregular equality constraints

Combining results of Avakov about tangent directions to equality constraints given by smooth operators with results of Ben-Tal and Zowe, we formulate a second-order theory for optimality in the sense of Dubovitskii-Milyutin which gives nontrivial conditions also in the case of equality constraints given by nonregular operators. Secondorder feasible and tangent directions are defined to construct conical approximations to inequality and equality constraints which within a single construction lead to first- and second-order conditions of optimality for the problem also in the nonregular case. The definitions of secondorder feasible and tangent directions given in this paper allow for reparametrizations of the approximating curves and give approximating sets which form cones. The main results of the paper are a theorem which states second-order necessary condition of optimality and several corollaries which treat special cases. In particular, the paper generalizes the Avakov result in the smooth case.This research was supported by NSF Grant DMS-91-009324, NSF Grant DMS-91-00043, SIUE Research Scholar Award and Fourth Quarter Fellowship, Summer 1992.     

Second order sufficient conditions for optimizing with equality constraints

The paper establishes the equivalence to second order of the Jacobean (constrained gradient) and the Lagrangean approaches to optimizing a function subject to equality constraints. The second order equivalence does not seem to have been considered previously. It is established using a new second order sufficiency condition which has computational advantages over the other two.     

Locally Lipschitz vector optimization with inequality and equality constraints

The present paper studies the following constrained vector optimization problem: \(\mathop {\min }\limits_C f(x),g(x) \in - K,h(x) = 0\), where f: ? ⁿ → ? ^m , g: ? ⁿ → ? ^p are locally Lipschitz functions, h: ? ⁿ → ? ^q is C ¹ function, and C ? ? ^m and K ? ? ^p are closed convex cones. Two types of solutions are important for the consideration, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point x ⁰ to be a w-minimizer and first-order sufficient conditions for x ⁰ to be an i-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done.     

On-line optimization design of sliding mode guidance law with multiple constraints

The design of terminal guidance law with impact angle constraint is required for air-to-ground guided weapons to increase their warhead effect. The variable structure guidance law that consists of diving plane guidance and turning plane guidance equations with impact angle constraint is derived, and the saturation function is introduced into the design of reaching law control to weaken the chattering of the guidance system. The influence of four guidance parameters (i.e., reaching law factor, switching item gain, angle error item factor, and boundary layer thickness) on guidance performance is studied and three typical constraints (i.e., heating rate, normal load factor, and dynamic pressure) are analyzed. An optimization model is established for this problem and the feasibility of on-line optimization on guidance law parameters by the Sequential Quadratic Programming (SQP) algorithm is discussed as well. Simulation results show that the on-line optimization of the derived guidance law not only satisfies specified constraints, but also minimizes the fuel cost during the flying course. Moreover, the optimization process can be completed in a few seconds so that it is suitable for on-board applications.     

Exact penalties for optimization problems with 2-regular equality constraints

A new first-order sufficient condition for penalty exactness that includes neither the standard constraint qualification requirement nor the second-order sufficient optimality condition is proposed for optimization problems with equality constraints.     

Global optimization of a quadratic functional with quadratic equality constraints

In this paper, we investigate a constrained optimization problem with a quadratic cost functional and two quadratic equality constraints. While it is obvious that, for a nonempty constraint set, there exists a global minimum cost, a method to determine if a given local solution yields the global minimum cost has not been established. We develop a necessary and sufficient condition that will guarantee that solutions of the optimization problem yield the global minimum cost. This constrained optimization problem occurs naturally in the computation of the phase margin for multivariable control systems. Our results guarantee that numerical routines can be developed that will converge to the global solution for the phase margin.     

A saddle-point criterion for convex problems with infinite-dimensional equality constraints

In this paper, we develop a saddle-point criterion for convex optimization problems with infinite-dimensional equality constraints. The method used in the derivation of this criterion is based on the property of openness of the equality operator. As an application, we develop necessary and sufficient conditions for an optimal control problem under a suitable controllability assumption. Constructing an optimal solution for a production planning problem is used as an illustrative example.The authors would like to thank Professor R. T. Rockafellar for his suggestions which have resulted in an improved version of the paper.     

Neighboring extremals of dynamic optimization problems with path equality constraints

Neighboring extremals of dynamic optimization problems with path equality constraints and with an unknown parameter vector are considered in this paper. With some simplifications, the problem is reduced to solving a linear, time-varying two-point boundary-value problem with integral path equality constraints. A modified backward sweep method is used to solve this problem. Two example problems are solved to illustrate the validity and usefulness of the solution technique. This research was supported in part by the National Aeronautics and Space Administration under NASA Grant No. NCC-2-106. The author is indebted to Professor A. E. Bryson, Jr., Department of Aeronautics and Astronautics, Stanford University, for many stimulating discussions.     

A primal-dual Newton-type algorithm for geometric programs with equality constraints

An augmented Lagrangian algorithm is used to find local solutions of geometric programming problems with equality constraints (GPE). The algorithm is Newton's method for unconstrained minimization. The complexity of the algorithm is defined by the number of multiplications and divisions. By analyzing the algorithm we obtain results about the influence of each parameter in the GPE problem on the complexity of an iteration. An attempt is made to estimate the number of iterations needed for convergence. In practice, certain hypotheses are tested, such as the influence of the penalty coefficient update formula, the distance of the starting point from the optimum, and the stopping criterion. For these tests, a random problem generator was constructed, many problems were run, and the results were analyzed by statistical methods.The authors are grateful to Dr. J. Moré, Argonne National Laboratory for his valuable comments.This research was partially funded by the Fund for the Advancement of Research at the Technion and by the Applied Mathematical Sciences Research Program (KC-04-02), Office of Energy Research, US Department of Energy, Contract No. W-31-109-Eng-38.     

Numerical solution of linear least-squares problems with linear equality constraints

In Ref. 1, a perturbation theory for the linear least-squares problem with linear equality constraints is presented. In this paper, the condition numbers of a general formula given in Ref. 1 are examined in order to compare them with the condition numbers of the two matrices of the problem. A class of test problems is also defined to study experimentally the numerical stability of three algorithms.This work was partially supported by the Ministero della Pubblica Istruzione, Rome, Italy.The authors thank the Centro Interdipartimentale di Calcolo Automatico e di Informatica Applicata of the University of Modena for having provided computing time on its VAX computer.     

On the perturbation of least squares problems with equality constraints

Some new perturbation results are presented for least squares problems with equality constraints, in which relative errors are obtained on perturbed solutions, least squares residuals, and vectors of Lagrange multipliers of the problem, based on the equivalence of the problem to a usual least squares problem and optimal perturbation results for orthogonal projections.