By Larranaga J. C. G.
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Extra resources for 3-Manifolds which are unions of three solid tori
However, even with this goal in mind, the level of exposition we offer will vary from topic to topic. This is deliberate. Some of the material we review is so basic that we feel it inappropriate that we should more than summarize what is known. If the reader feels we are going too fast, we provide references to the books and papers where this material is covered in considerably more depth. The remainder of the material is more directly relevant, and in this case we try to motivate and develop it so that we hope the reader need look no further.
1 Suppose that / e C1, and that x* is a local minimizer of /(x). 1) is known as the first-order (necessary) optimality condition for an unconstrained minimizer of /(x). 1) is said to be a first-order critical or first-order stationary point of /. We now turn to higher order derivatives. We shall use the following blanket assumption throughout the book. l holds and that x* is a local minimizer of /(x). 2) are known as the second-order (necessary) optimality conditions for an unconstrained minimizer of /(re), and any point x* that satisfies both conditions is said to be a second-order critical point of /.
Finite-valued convex functions defined on open sets are Lipschitz continuous, and their subdifferential and generalized gradient coincide. In general, we say that a function is regular at x if fd(x) exists for all d and f'd(x] = f d ( x ] . For Lipschitz continuous functions, the requirement is simply that one-sided and generalized directional derivatives coincide. The function is regular on X if it is regular for each x in X. A most important special case results when h is a regular, locally Lipschitz continuous function on a subset C of R m , and / and c are continuously differentiable functions from X C R n to R and C, respectively.