# A Course on Optimization and Best Approximation by Richard B. Holmes (auth.) By Richard B. Holmes (auth.)

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F) We now briefly analysis. if -~Yi course here the regularity sensitivity exists and then e) Exa___mple. The ordinary convex program defined by f(x) = Xl, a Of assumption is violated. consider the use of Lagrange multipliers This use is to provide an estimate in for the 40 change in value of an ordinary convex program when the constraint bounds are perturbed. The basic result, which depends on the Theorem in d), is the following. Theorem, as in 12c). ,fn For given inf {f(-): define an ordinary Y, F ~ Rn let x, ~ e X fi(.

X ~ K} ~ sup < K , y ~ function Let f = 6K on f(x) is a subspace = jJxll. of X, Then and f* = 6U(X, ). f(x) = dist (x,M), 44 then f* = 6U(Mi ). 3) Then Let f*(Y) 4) X be a nls and for = I]yl ]q/q, Let X = R1 where and f(x) = e x n n × Let X = Rn matrix. If +~ 0 = and let A i/p + i/q = 1 f(x) f(x) = llxllP/p. Then for for y < 0 y = 0 y(log y-l) s) let i/p + I/q = i. i f*(y) 1 < p < +~ for y > 0. be a symmetric positive definite and (1/p)(x,Ax>P/2, = then f*(y) Exercise 23. d) Theorem. = (l/q) < y , A - l y > q/2 Verify these examples.

Namely, is bounded above on some non-empty open set, in particular, upper semi-continuous (usc) at any point, continuous int throughout (dom (f)) f ~ Conv continuous (X) throughout where rel-int X (dom (f)). ,~n) II = f As a special case it follows (dom (f)) is an open f(Xo+X)-f(Xo) f is necessarily is finite dimensional, last assertion is also available. assume that f if if [4, p. 92; 42, p. 193]. special case of this was proved in 3b)). that if then to - f(Xo) R n. small, 29 which tends interval to must Now since we 0 as x ÷ 8, be continuous.