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chapter3 - 3 EXISTENCE OF GLOBAL OPTIMIZERS This chapter...

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3. EXISTENCE OF GLOBAL OPTIMIZERS This chapter presents theoretical tools for demonstrating the existence of global minimizers or maximizers. For practical purposes, such tools enable us to verify global optimality simply by comparing the function values of all candidates for optimality. 3.1 Extreme Value Theory Our first result underlies most of the existence theory in optimization. Theorem 3.1.1 (Weierstrass) . Suppose that S R n is nonempty, closed and bounded. Then every continuous function on S attains a global maximum and a global minimum. Proof. Deferred until later in this section. We refer the reader to section 1.4 for definitions and examples of the terms “closed,” “bounded,” and “continuous.” The following special case of the Weierstrass theorem is usually introduced in the first Calculus course. Corollary 3.1.2 (One-dimensional extreme value theorem) . If f : [ a, b ] R is continuous then f attains a global maximum and a global minimum on [ a, b ] . The extreme value theorem provides information even in seemingly more general cir- cumstances, as shown by the next result. Corollary 3.1.3 (Existence via one-dimensional sublevel boundedness) . Let I be any interval in R . Suppose f : I R is continuous and there is some number c I so that the following are satisfied (as applicable): (a) lim x a + f ( x ) > f ( c ) , if a is the left endpoint of I ; (b) lim x →-∞ f ( x ) > f ( c ) , if I has no left endpoint; (c) lim x b - f ( x ) > f ( c ) , if b is the right endpoint of I ; (d) lim x →∞ f ( x ) > f ( c ) , if I has no right endpoint. Then f has a global minimizer on I . Sketch of Proof. A closed and bounded interval is hiding in the definition of the limit! For example, “lim x a + f ( x ) = d means “ ² > 0, δ > 0 , x ( a, a + δ ) : | f ( x ) - d | < ² .” “lim x a + f ( x ) = means “ M, δ > 0 , x ( a, a + δ ) : f ( x ) > M .” “lim x →-∞ f ( x ) = ” means “ M, N, x ( -∞ , N ) : f ( x ) > M .” “lim x →-∞ f ( x ) = d means “ ² > 0 , N, x ( -∞ , N ) : | f ( x ) - d | < ² .” 29
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Conditions (a)–(b) grant the existence of a left endpoint α < c and conditions (c)–(d) give a right endpoint β > c so that all values of x outside of [ α, β ] yield f ( x ) > c . We therefore can restrict our search to this nonempty, closed, bounded interval [ α, β ]. Example 3.1.4 (Using sublevel boundedness) . The function f ( x ) = ( x 4 - 6 x 3 + 3 x 2 - x + 15) / (100 - x 2 ) has a global minimizer on the open interval ( - 10 , 10) because lim x →- 10 + f ( x ) = = lim x 10 - f ( x ) . Observation 3.1.5 (General use of one-dimensional “coercivity”) . Combined with the critical point test, knowledge of the existence of a global minimizer in an interval I allows us to compare the values of f at the following candidates: (a) critical points of f in I ; (b) the endpoints (if any) of I ; (c) those points in I where f 0 does not exist. The global minimizers for f on I are the candidates giving the smallest value for f x ).
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