{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

chapter3

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern