FRONTIERS IN OPTIMIZATION AND CONTROL S.H. Hou, X.M. Yang and G.Y. Chen (eds)

1 METHODS IN OPTIMIZATION

author

Department of AAA

Abstract: It is known that optimization is useful ...

Key words: optimization theory, optimization method.

1 INTRODUCTION Optimization has many applications in finance and engineering in e.g., (Bertsekas (1976)).... 2 AN OPTIMIZATION PROBLEM 2.1 this is subsection title Consider the following constrained optimization problem (P):

inf_{x∈X}  f_{0}(x)
subject to  f_{i}^{2}(x) ≤ 0, i = 1, ⋯, m,

where X ⊂ R^{n} is a subset of R^{n}, f_{i} : X → R, i = 0, 1, 2, ⋯, m are real-valued functions.

Definition 2.1 The Lagrangian function of (P) is defined as

L(x, λ) = f_{0}(x) + ∑_{i=1}^{m} λ_{i}f_{i}(x).  (2.1)

Figure 3.1 Short caption.

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Lemma 2.1 If all f_{i} are convex, then the Lagrangian is convex.

Proof: Follows directly from the definition of ϕ.

Theorem 2.1 (Weak duality)

f_{0}(x) ≥ 0.  (2.2)

It is also not difficult to see that an equality constraint, if any, can be easily converted into two inequality constraints. Thus our results are applicable to optimization problems with both inequality and equality constrained optimization problems, see Example 1.

Next we consider the case when X is a discrete set but has an infinite number of elements. The proof of next result is similar to that of Theorem ??, see also (??) with some modification to allow for the case that X is now a discrete set but has an infinite number of elements. In Bertsekas (1976) the case that X is a connected set was considered.

1. This is the first item in the numbered list.

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3 ALL THE THINGS THAT CAN BE DONE WITH FIGURE CAPTIONS

GLOBAL OPTIMIZATION 3

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Figure 3.6a Lettered caption.

Figure 3.6b One caption.  Figure 3.6c Two captions.

4 MAKING TABLES

Notice that the caption should be at the top of the table. Use a line above the table, under the column heads, and at the end of the table. This form of the tabular command makes the table spread out to the width of the page.

Table 4.1 Effects of the Two Types of Scaling Proposed by Dennard and Co-Workers.^{a,b}

Parameter  κ Scaling  κ, λ Scaling
Dimension  κ^{−1}  λ^{−1}
Voltage  κ^{−1}  κ^{−1}
Currant  κ^{−1}  λ/κ^{2}
Dopant Concentration  κ  λ^{2}/κ

aRefs. 19 and 20. bκ, λ > 1.

Table 4.2a A small table with a lettered table caption.

αβΓΔ One Two Three
one  two three
one  two three

Table 4.3 Here is a table caption.

Cell Time (sec.)
1  432.22
2  32.32
3  2.32

Table 4.3 (continued) This is a continued caption.

Cell Time (sec.)
4  532.22
5  12.02
6  4.44

5 CONCLUSIONS In this sample file, the requirements on how to prepare the book chapter using the Kluwer style file were given.

Acknowledgments

The authors wish to thank Drs. X, Y and Z for their encouragement and support.

Appendix

This is an appendix without a title.

References

Ahuja, R.K., Magnanti, T.L. and Orlin, J.B. (1993), Network flows, Prentice

Hall, Englwood Cliffs, New Jersey. Bertsekas, D.P. (1976), Nondifferentiable optimization via approximation, Math-

ematical Programming, Vol. 3, pp. 1-25.