Consider a maximization problem and an iterative algorithm for solving it. The algorithm generates a sequence of points increasing in the value of the objective function. Two cases may occur for the sequence of generated points: (1) The sequence converges to a point or has no cluster point. (2) The sequence has a cluster point but does not converge to the duster point.It is a natural viewpoint that the sequence in case (1) is easier to be handled than that in case (2). Thus, in the literature, when the global convergence of an algorithm is hard to be established, one usually first study it with assumption that the generated sequence is convergent.

 This volume is monograph.It covers various aspects of convergence theory about feasible direction methods innonlinear programming.Many basic results inclnded here have not been contained by other books.It provides a useful state-of-art review of activity in these field.

Chapter 1. Introduction and Preliminaries

 1.1. Introduction

 1.2. Basic Notations and Terminologies

 1.3. Optimality Conditions

 1.4. Line Search

 1.5. A Remark

Chapter 2. Slope Lemmas

 2.1. The First Slope Lemma

 2.2. An Example

 2.3. Gradient Projection on Tangent Plane

 2.4. The Second Slope Lemma

Chapter 3. Rosen's Method and Its Convergence

 3.1. Rosen's Method

 3.2. Choice of Parameter

 3.3 Global Convergence

 3.4. Rate of Convergence

 3.5. Kantorovich Inequality

 3.6. Degeneracy

Chapter 4. Combining with Variable Metric Methods

 4.1. A Variation of Goldfarb's Method

 4.2. Pu-Yu's Theorem

Chapter 5. e-Active Set Strategy

 5.1. Decomposition of Polyhedron

 5.2. Stability of Regularity (I)

 5.3. An Example

 5.4 Stability of Regularity (II)

 5.5. Rotating Tangent Plane

Chapter 6. Reduced Gradient Methods

 6.1 Wolfe's Method

 6.2. Nondegeneracy Assumption

 6.3. Yue-Han's Pivot

 6.4. Convergence of Wolfe's Method

 6.5. Luenberger's and Wang's Variations

 6.6. Improving Rosen's Method

Chapter 7. Point-to-Set Mapping

 7.1. Zangwill's Theorem

 7.2. Consequences of Slope Lemmas

 7.3. Comparison

Chapter 8. On Other Topics

 8.1. The Third Slope Lemma

 8.2. Ritter's Rule and Its Extension

 8.3. Global Convergence Rate

 8.4. Karmarkar's Alogrithm

 8.5. Nonsmooth Optimization

References