This book presents a comprehensive treatment of necessary conditions for general optimization problems. The presentation is carried out in the context of a general theory for extremal problems in a topological vector space setting.
Following a brief summary of the required background, generalized Lagrange multiplier rules are derived for optimization problems with equality and generalized "inequality" constraints. The treatment stresses the importance of the choice of the underlying set over which the optimization is to be performed, the delicate balance between differentiabilitycontinuity requirements on the constraint functionals, and the manner in which the underlying set is approximated by a convex set. The generalized multiplier rules are used to derive abstract maximum principles for classes of optimization problems defined in terms of operator equations in a Banach space. It is shown that special cases include the usual maximum principles for general optimal control problems described in terms of diverse systems such as ordinary differential equations, functional differential equations, Volterra integral equations, and difference equations. Careful distinction is made throughout the analysis between "local" and "global" maximum principles.
Originally published in 1977.
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Front Matter Front Matter (pp. ivi) 
Table of Contents Table of Contents (pp. viiviii) 
PREFACE PREFACE (pp. ixxii)Η. T. Banks 
SUMMARY OF NOTATION SUMMARY OF NOTATION (pp. xiii2) 
CHAPTER I Mathematical Preliminaries CHAPTER I Mathematical Preliminaries (pp. 362)In this chapter we shall present some mathematical background material which will be necessary in order to understand the remainder of this book. For the most part, we shall confine ourselves to stating definitions and to theorem statements. The proofs of many of the theorems are immediate. References will be given for those proofs which are less obvious but which are widely available in the literature. (In this regard, the reference numbers refer to books listed at the end of this chapter.) Proofs will be given for those lemmas and theorems which either are novel or are not widely known....

CHAPTER II A Basic Optimization Problem in Simplified Form CHAPTER II A Basic Optimization Problem in Simplified Form (pp. 6395)This chapter is in many ways a capsule version of the entire book. We shall here investigate the optimization problem which consists in finding the minimum of a functionϕ^{0}on a given set
${\cal {E}}$ , subject to equality constraintsφ^{i}(e) = 0 fori= 1,…,mand inequality constraintsϕ^{i}(e) ≤ 0 fori= 1,…,μ, whereφ^{1},…,φ^{m},ϕ^{0},ϕ^{1},…,ϕ^{μ}are given, realvalued functions defined on${\cal {E}}$ . Under the assumptions that${\cal {E}}$ is a subset of a linear vector space which is finitely open in itself [see (I.1.39)], thatφ^{1},…,φ^{m}are finitely differentiable... 
CHAPTER III A General Multiplier Rule CHAPTER III A General Multiplier Rule (pp. 96143)Our aim in this chapter is to generalize the multiplier rules [Theorems (II.1.4 and 18)] developed in Section 1 of the preceding chapter. Our generalizations will be in a number of directions. First, we shall weaken the requirement that the underlying set
${\cal {E}}$ is finitely open in itself, in order to be able to apply our results to more general optimization problems with operator equation constraints than those considered in Chapter II [as was discussed in the introduction to Chapter II and in (II.2.47)]. Second, we shall allow optimization problems with inequality constraints that are considerably more general than those... 
CHAPTER IV Optimization with Operator Equation Restrictions CHAPTER IV Optimization with Operator Equation Restrictions (pp. 144211)In this chapter, we shall justify the form of the hypotheses under which we derived the various multiplier rules in the preceding chapter [namely, see Hypotheses (III.2.11–14 and 57–59)].
We shall be concerned with optimization problems (ordinary, minimax, and with vectorvalued criterion function) in which we are given a Banach space
${\cal {X}}$ , an open subsetAof${\cal {X}}$ , and a set${\cal {W}}$ of continuously differentiable operators$T:A\to {\cal {X}}$ , and we wish to find a functionx∈Awhich satisfies the equationx=Tx, for some$T\in {\cal {W}}$ , as well as some prescribed equality and... 
CHAPTER V Optimal Control Problems with Ordinary Differential Equation Constraints CHAPTER V Optimal Control Problems with Ordinary Differential Equation Constraints (pp. 212262)In this chapter we shall carry out a detailed investigation of some particular cases of the general optimization problems which were discussed in Chapter IV. Namely, we shall study the case where the operator equationsx=TxwithT∈Wrepresent ordinary differential equations with a socalled control variable.
1 More precisely, we shall concern ourselves with problems which are defined in terms of some givennvectorvalued functionf, whose domain is G ×V×I, where G is a given open set inR^{n},Vis a given arbitrary set, andIis a given compact...

CHAPTER VI Optimal Control Problems with Parameters and Related Problems CHAPTER VI Optimal Control Problems with Parameters and Related Problems (pp. 263314)In this chapter, we shall continue our investigation of optimal control problems with ordinary differential equation constraints. But now we shall exclusively be concerned with problems that can be expressed in parametric form, using the viewpoint of Section 2 of Chapter IV. This “parameter” will, in much (though not all) of this chapter turn out to be the control function.
In Section 1, we shall examine a freetime optimal control problem which is in the spirit of Problem (V.2.1), but in which the functionfin the righthand side of the differential equation, together with the functionsχ^{ι}(which define...

CHAPTER VII Miscellaneous Optimal Control Problems CHAPTER VII Miscellaneous Optimal Control Problems (pp. 315360)In the preceding two chapters, we have been concerned exclusively with optimal control problems with restrictions in the form of ordinary differential equations. However, the machinery which we developed in Chapter IV allows us to consider much more general classes of restrictions, such as operator equations in terms of Volterratype operators [see (IV.3.3)].
We shall illustrate the way in which we can generalize the ordinary differential equation restrictions by considering two generalizations of such equations: functional differential equations (in Section 1)—including ordinary differential equations with retardations as a special case (in Section 2), and Volterra integral equations (in Section...

APPENDIX: VolterraType Operators APPENDIX: VolterraType Operators (pp. 361383) 
NOTES AND HISTORICAL COMMENTS NOTES AND HISTORICAL COMMENTS (pp. 384412) 
REFERENCES REFERENCES (pp. 413421) 
SUBJECT INDEX SUBJECT INDEX (pp. 422424) 
Back Matter Back Matter (pp. 425425)