The Conjugate Gradient Method 9. The Partial Conjugate Gradient Method 9. Extension to Nonquadratic Problems 9. Parallel Tangents 9. Exercises Chapter Quasi-Newton Methods Modified Newton Method Construction of the Inverse Davidon—Fletcher—Powell Method The Broyden Family Convergence Properties Scaling Summary Constrained Minimization Conditions Constraints Tangent Plane Examples Second-Order Conditions Eigenvalues in Tangent Subspace Sensitivity Inequality Constraints Zero-Order Conditions and Lagrange Multipliers Primal Methods Advantage of Primal Methods Feasible Direction Methods Active Set Methods The Gradient Projection Method Convergence Rate of the Gradient Projection Method The Reduced Gradient Method Convergence Rate of the Reduced Gradient Method Variations Penalty and Barrier Methods Penalty Methods Barrier Methods Properties of Penalty and Barrier Functions Conjugate Gradients and Penalty Methods Normalization of Penalty Functions Penalty Functions and Gradient Projection Exact Penalty Functions Dual and Cutting Plane Methods Global Duality Local Duality Dual Canonical Convergence Rate Separable Problems Augmented Lagrangians The Dual Viewpoint Cutting Plane Methods Modifications Primal-Dual Methods The Standard Problem Strategies A Simple Merit Function Basic Primal—Dual Methods Modified Newton Methods Descent Properties Rate of Convergence Interior Point Methods Semidefinite Programming Exercises Appendix A.

Mathematical Review A. Sets A. Matrix Notation A. Spaces Contents xiii A. Eigenvalues and Quadratic Forms A.

## Local Optimization Software

Topological Concepts A. Functions Appendix B. Convex Sets B. Basic Definitions B. Hyperplanes and Polytopes B. Separating and Supporting Hyperplanes B. Extreme Points Appendix C. Gaussian Elimination Bibliography Index It offers a certain degree of philosophical elegance that is hard to dispute, and it often offers an indispensable degree of operational simplicity.

Using this optimization philosophy, one approaches a complex decision problem, involving the selection of values for a number of interrelated variables, by focussing attention on a single objective designed to quantify performance and measure the quality of the decision. This one objective is maximized or minimized, depending on the formulation subject to the constraints that may limit the selection of decision variable values.

If a suitable single aspect of a problem can be isolated and characterized by an objective, be it profit or loss in a business setting, speed or distance in a physical problem, expected return in the environment of risky investments, or social welfare in the context of government planning, optimization may provide a suitable framework for analysis.

It is, of course, a rare situation in which it is possible to fully represent all the complexities of variable interactions, constraints, and appropriate objectives when faced with a complex decision problem. Thus, as with all quantitative techniques of analysis, a particular optimization formulation should be regarded only as an approximation. Skill in modelling, to capture the essential elements of a problem, and good judgment in the interpretation of results are required to obtain meaningful conclusions. Optimization, then, should be regarded as a tool of conceptualization and analysis rather than as a principle yielding the philosophically correct solution.

Skill and good judgment, with respect to problem formulation and interpretation of results, is enhanced through concrete practical experience and a thorough under- standing of relevant theory. Problem formulation itself always involves a tradeoff between the conflicting objectives of building a mathematical model sufficiently complex to accurately capture the problem description and building a model that is tractable.

The expert model builder is facile with both aspects of this tradeoff. One aspiring to become such an expert must learn to identify and capture the important issues of a problem mainly through example and experience; one must learn to distinguish tractable models from nontractable ones through a study of available technique and theory and by nurturing the capability to extend existing theory to new situations.

Examples of situations leading to this structure are sprinkled throughout the book, and these examples should help to indicate how practical problems can be often fruitfully structured in this form.

### Linear Programming

The book mainly, however, is concerned with the development, analysis, and comparison of algorithms for solving general subclasses of optimization problems. The last two parts together comprise the subject of nonlinear programming. Linear Programming Linear programming is without doubt the most natural mechanism for formulating a vast array of problems with modest effort. A linear programming problem is charac- terized, as the name implies, by linear functions of the unknowns; the objective is linear in the unknowns, and the constraints are linear equalities or linear inequal- ities in the unknowns.

## Nonlinear programming

One familiar with other branches of linear mathematics might suspect, initially, that linear programming formulations are popular because the mathematics is nicer, the theory is richer, and the computation simpler for linear problems than for nonlinear ones. But, in fact, these are not the primary reasons.

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In terms of mathematical and computational properties, there are much broader classes of optimization problems than linear programming problems that have elegant and potent theories and for which effective algorithms are available. It seems that the popularity of linear programming lies primarily with the formulation phase of analysis rather than the solution phase—and for good cause. For one thing, a great number of constraints and objectives that arise in practice are indisputably linear.

The linearity of the budget constraint is extremely natural in this case and does not represent simply an approximation to a more general functional form. Another reason that linear forms for constraints and objectives are so popular in problem formulation is that they are often the least difficult to define. Thus, even if an objective function is not purely linear by virtue of its inherent definition as in the above example , it is often far easier to define it as being linear than to decide on some other functional form and convince others that the more complex form is the best possible choice.

Linearity, therefore, by virtue of its simplicity, often is selected as the easy way out or, when seeking generality, as the only functional form that will be equally applicable or nonapplicable in a class of similar problems.

Of course, the theoretical and computational aspects do take on a somewhat special character for linear programming problems—the most significant devel- opment being the simplex method. This algorithm is developed in Chapters 2 and 3. More recent interior point methods are nonlinear in character and these are developed in Chapter 5. Unconstrained Problems It may seem that unconstrained optimization problems are so devoid of struc- tural properties as to preclude their applicability as useful models of meaningful problems. Quite the contrary is true for two reasons. First, it can be argued, quite convincingly, that if the scope of a problem is broadened to the consideration of all relevant decision variables, there may then be no constraints—or put another way, constraints represent artificial delimitations of scope, and when the scope is broadened the constraints vanish.

Thus, for example, it may be argued that a budget constraint is not characteristic of a meaningful problem formulation; since by borrowing at some interest rate it is always possible to obtain additional funds, and hence rather than introducing a budget constraint, a term reflecting the cost of funds should be incorporated into the objective.

A similar argument applies to constraints describing the availability of other resources which at some cost however great could be supplemented. The second reason that many important problems can be regarded as having no constraints is that constrained problems are sometimes easily converted to uncon- strained problems.

For instance, the sole effect of equality constraints is simply to limit the degrees of freedom, by essentially making some variables functions of others. These dependencies can sometimes be explicitly characterized, and a new problem having its number of variables equal to the true degree of freedom can be determined. Aside from representing a significant class of practical problems, the study of unconstrained problems, of course, provides a stepping stone toward the more general case of constrained problems.

Many aspects of both theory and algorithms Constrained Problems In spite of the arguments given above, many problems met in practice are formulated as constrained problems. This is because in most instances a complex problem such as, for example, the detailed production policy of a giant corporation, the planning of a large government agency, or even the design of a complex device cannot be directly treated in its entirety accounting for all possible choices, but instead must be decomposed into separate subproblems—each subproblem having constraints that are imposed to restrict its scope.

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## Nonlinear Programming - MATLAB & Simulink

Thus, in a planning problem, budget constraints are commonly imposed in order to decouple that one problem from a more global one. Therefore, one frequently encounters general nonlinear constrained mathematical programming problems. The set S is a subset of n-dimensional space. The function f is the objective function of the problem and the equations, inequalities, and set restrictions are constraints. Generally, in this book, additional assumptions are introduced in order to make the problem smooth in some suitable sense. For example, the functions in the problem are usually required to be continuous, or perhaps to have continuous derivatives.

This ensures that small changes in x lead to small changes in other values associated with the problem. Also, the set S is not allowed to be arbitrary but usually is required to be a connected region of n-dimensional space, rather than, for example, a set of distinct isolated points. This ensures that small changes in x can be made. Indeed, in a majority of problems treated, the set S is taken to be the entire space; there is no set restriction.

In view of these smoothness assumptions, one might characterize the problems treated in this book as continuous variable programming, since we generally discuss problems where all variables and function values can be varied continuously. In fact, this assumption forms the basis of many of the algorithms discussed, which operate essentially by making a series of small movements in the unknown x vector.

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As might be expected, the size of problems that can be effectively solved has been increasing with advancing computing technology and with advancing theory. Today, with present computing capabilities, however, it is reasonable to distinguish three classes of problems: small-scale problems having about five or fewer unknowns and constraints; intermediate-scale problems having from about five to a hundred or a thousand variables; and large-scale problems having perhaps thousands or even millions of variables and constraints. This classification is not entirely rigid, but it reflects at least roughly not only size but the basic differences in approach that accompany different size problems.

As a rough rule, small-scale problems can be solved by hand or by a small computer. Intermediate-scale problems can be solved on a personal computer with general purpose mathematical programming codes. Large-scale problems require sophisticated codes that exploit special structure and usually require large computers.