Constrained Optimization Problems

The general constrained optimization problem can be considered as minimizing a function of n variables F(x), subject to a series of m constraints of the fomi C.(x) = 0. In the penalty fiinction method, additional temis of the fomi. (x), a.> 0, are fomially added to the original fiinction, thus... 

Each of the inequality constraints gj(z) multiphed by what is called a Kuhn-Tucker multiplier is added to form the Lagrange function. The necessaiy conditions for optimality, called the Karush-Kuhn-Tucker conditions for inequality-constrained optimization problems, are... 

Further Comments on General Programming.—This section will utilize ideas developed in linear programming. The use of Lagrange multipliers provides one method for solving constrained optimization problems in which the constraints are given as equalities. 

Slightly different constraints are used to illustrate the mathematical technique. In this example, the constrained optimization problem is to locate levels of stearic acid (X ) and starch (X2) that minimize the time of in vitro release (y2) such that the average tablet volume (jy) did not exceed 9.422 cm2 and the average friability (y3) did not exceed 2.72%. 

An important class of the constrained optimization problems is one in which the objective function, equality constraints and inequality constraints are all linear. A linear function is one in which the dependent variables appear only to the first power. For example, a linear function of two variables x and x2 would be of the general form ... 

Techniques for unconstrained and constrained optimization problems generally involve repeated use of a one-dimensional search as described in Chapters 6 and 8. 

EXAMPLE 8.1 GRAPHIC INTERPRETATION OF A CONSTRAINED OPTIMIZATION PROBLEM... 

Geometry of a constrained optimization problem. The feasible region lies within the binding constraints plus the boundaries themselves. 

Like penalty methods, barrier methods convert a constrained optimization problem into a series of unconstrained ones. The optimal solutions to these unconstrained subproblems are in the interior of the feasible region, and they converge to the constrained solution as a positive barrier parameter approaches zero. This approach contrasts with the behavior of penalty methods, whose unconstrained subproblem solutions converge from outside the feasible region. 

Fogel, D. B. A Comparison of Evolutionary Programming and Genetic Algorithms on Selected Constrained Optimization Problems. Simulation 64 397-404 (1995). 

For measurement adjustment, a constrained optimization problem with model equations as constraints is resolved at a fixed interval. In this context, variable classification is applied to reduce the set of constraints, by eliminating the unmeasured variables and the nonredundant measurements. The dimensional reduction of the set of constraints allows an easier and quicker mathematical resolution of the problem. 

The operation of a plant under steady-state conditions is commonly represented by a non-linear system of algebraic equations. It is made up of energy and mass balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem. 

In many crystallographic problems, the choice of the variables x is subject to constraints (boundary conditions represented by equations). The problem is then known as a constrained optimization problem. An example would be the refinement of a... 

A similar constrained optimization problem has been solved in Section 2.5.4 by the method of Lagrange multipliers. Using the same method we look for the stationary point of the Lagrange function... 

A key idea in developing necessary and sufficient optimality conditions for nonlinear constrained optimization problems is to transform them into unconstrained problems and apply the optimality conditions discussed in Section 3.1 for the determination of the stationary points of the unconstrained function. One such transformation involves the introduction of an auxiliary function, called the Lagrange function L(x,A, p), defined as... 

An important class of constrained optimization problems is one in which both the objective function and the constraints are linear. The solution of these problems is highly structured and can be obtained... 

In addition to a wide variety of problem types, there are three common types of constrained optimization problems that are typically of interest linear programs (LPs), quadratic programs (QPs), and nonlinear programs (NLPs). 

Algorithm for Constrained Optimization Problems", presented at Biennial Numerical Analysis Conference, Dundee, 1979. 

Gradient Norm Minimizations 14.5.9 Netvton-Raphson Methods 14.5.10 Gradient Extremal Methods Constrained Optimization Problems Locating the Global Minimum and Conformational Sampling 333 333 338 338 339 Appendix C First and Second Quantization Reference 411 411 412... 

A typical SQP termination condition for a constrained optimization problem... 

In summary, condition 1 gives a set of n algebraic equations, and conditions 2 and 3 give a set of m constraint equations. The inequality constraints are converted to equalities using h slack variables. A total of M + m constraint equations are solved for n variables and m Lagrange multipliers that must satisfy the constraint qualification. Condition 4 determines the value of the h slack variables. This theorem gives an indirect problem in which a set of algebraic equations is solved for the optimum of a constrained optimization problem. 

In this case the chemical reaction equilibrium problem is expressed so that we are minimizing the free energy directly as formally defined by the fundamental statement (6.37). In mathematical terms (6.37) represents a constrained optimization problem. This type of problems is usually solved by the use of Lagrange multipliers. 

Economists, such as Hildenbrand( 3), who study physical production processes also fix their gazes upon a cost minimization objective but they assume that the engineer has already solved his constrained optimization problem. The economic theory of cost and production describes the effects of variable input prices upon cost-minimizing combinations of material and nonmaterial inputs. 

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