By Wenyu Sun
This ebook, end result of the the authors’ educating and examine event in a variety of universities and institutes during the last ten years, can be utilized as a textbook for an optimization path for graduates and senior undergraduates. It systematically describes optimization concept and a number of other strong equipment, together with contemporary effects. for many equipment, the authors speak about an idea’s motivation, examine the derivation, determine the worldwide and native convergence, describe algorithmic steps, and speak about the numerical functionality. The e-book bargains with either thought and algorithms of optimization at the same time. It additionally includes an in depth bibliography with 366 references. eventually, except its use for instructing, Optimization conception and Methods is additionally very helpful for doing study.Audience
This booklet is meant for senior scholars, graduates, lecturers, and researchers in optimization, operations examine, computational arithmetic, utilized arithmetic, and a few engineering and economics. it is going to even be invaluable for scientists in engineering and economics.
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Additional info for Optimization Theory and Methods: Nonlinear Programming
Example text
Let x1 , x2 ∈ S and (x1 , α1 ), (x2 , α2 ) be in epif . 7 that f (λx1 + (1 − λ)x2 ) ≤ λf (x1 ) + (1 − λ)f (x2 ) ≤ λα1 + (1 − λ)α2 for any λ ∈ (0, 1). Since S is a convex set, λx1 + (1 − λ)x2 ∈ S. Hence (λx1 + (1 − λ)x2 , λα1 + (1 − λ)α2 ) ∈ epi f , which means epif is convex. Conversely, assume that epif is convex, and let x1 , x2 ∈ S and (x1 , f (x1 )), (x2 , f (x2 )) ∈ epif . Then we have from the convexity of epif that (λx1 + (1 − λ)x2 , λf (x1 ) + (1 − λ)f (x2 )) ∈ epif , for λ ∈ (0, 1). 3.
R Similarly, for the negative rank-one update of Cholesky factorization , we have ¯ = B − yy T = L(D − ppT )LT B ˆD ˆL ˆ T LT = L ¯D ¯L ¯T . 75) ¯ become zero or Since, in this case, it is possible that the elements of D negative due to round-off error, this phenomenon must be taken into consideration. The following algorithm keeps all d¯j (j = 1, · · · , n) positive. 20 CHAPTER 1. 19 (Cholesky Factorization of Negative Rank-One Update) 1. Solve Lp = y for p. Set tn+1 = 1 − pT D−1 p. If tn+1 < M , set tn+1 = M , where M is the relative precision of the computer.
S ∗ is a closed convex cone. 2. S ⊂ S ∗∗ , where S ∗∗ is the polar cone of S ∗ . If S is a nonempty closed convex set, then S ∗∗ = S. 3. If S1 , S2 are nonempty sets, then S1 ⊂ S2 implies S2∗ ⊂ S1∗ . 36 CHAPTER 1. INTRODUCTION The normal and tangent cones play a special role in constrained optimization. Here we give their definitions below. Let S be a closed convex set. The normal cone of S at x ¯ is defined as ¯ ≤ 0, ∀x ∈ S}. 5) ¯), λ ≥ 0, x ∈ S}, = {d | d = lim λ(x − x x→¯ x where cl{S} denotes the closure of S.