Читать книгу Linear and Convex Optimization - Michael H. Veatch - Страница 4

1 Chapter 1Figure 1.1 Region satisfying constraints for sending aid.Figure 1.2 Optimal point and contour for sending aid.Figure 1.3 Problem has optimal solution for dashed objective but is unbounde...Figure 1.4 Feasible integer solutions for (1.5).

2 Chapter 2Figure 2.1 Electricity transmission network.Figure 2.2 Transportation network for soybeans.Figure 2.3 Water pipe network for Exercise 2.25.

3 Chapter 3Figure 3.1 Profit contribution (the negative of cost) for labor.Figure 3.2 Street grid. Each block is labeled with its travel time and each ...Figure 3.3 Travel times for Exercise 3.12.Figure 3.4 Project costs for Exercise 3.13.

4 Chapter 4Figure 4.1 A piecewise linear function.

5 Chapter 5Figure 5.1 An improving direction for maximizing .

6 Chapter 6Figure 6.1 Feasible region and isocontours for Example 6.1.Figure 6.2 The first set is convex. The second is not.Figure 6.3 The point is a convex combination of , , .Figure 6.4 Unbounded sets. Only the first two have unbounded directions.Figure 6.5 A polyhedral cone.Figure 6.6 The first function is convex. The second is concave.Figure 6.7 The line only intersects at the point shown.Figure 6.8 Epigraph of .

7 Chapter 7Figure 7.1 Basic solutions for Example 7.1.Figure 7.2 The point is a degenerate basic feasible solution.Figure 7.3 Feasible region for Example 7.3.Figure 7.4 Edge directions for the bfs .Figure 7.5 Cones and their extreme rays .

8 Chapter 8Figure 8.1 Gradient vectors and lines of constant .Figure 8.2 Gradient vectors and active constraint normal vectors.

9 Chapter 9Figure 9.1 A polygon with sides has diameter 4.

10 Chapter 10Figure 10.1 Feasible region for (10.1) with constraint .Figure 10.2 Feasible region for 10.1 with .Figure 10.3 Optimal value as a function of .Figure 10.4 Feasible regions with right‐hand sides .

11 Chapter 11Figure 11.1 Heavy lines are a spanning tree of the directed graph.Figure 11.2 A network flow problem for Figure 11.1.Figure 11.3 Three spanning tree solutions. The dotted line is an entering ar...Figure 11.4 An infeasible network flow problem.Figure 11.5 A transportation problem with three supply nodes and two demand ...Figure 11.6 Initial tree solution for Figure 11.5.Figure 11.7 Final tree solution for Figure 11.5.Figure 11.8 Progress of algorithm for Example 11.7.

12 Chapter 12Figure 12.1 Feasible region for Example 12.1.Figure 12.2 Feasible region and convex hull for Example 12.1.Figure 12.3 Renumbering a spanning tree. Node and arc numbers in parentheses...

13 Chapter 13Figure 13.1 A branch and bound tree.Figure 13.2 Branch and bound tree for Example 13.1 after the first branching...Figure 13.3 Branch and bound tree for Example 13.1.Figure 13.4 Branch and bound tree for Example 13.2.Figure 13.5 A cutting plane for .Figure 13.6 Feasible region for Example 13.4. The dashed line is a C–G inequ...

14 Chapter 14Figure 14.1 Optimality condition with one constraint .Figure 14.2 Optimality condition with one constraint .Figure 14.3 Optimality condition with two inequality constraints.Figure 14.4 Points and satisfy the KKT conditions but are not local maxi...Figure 14.5 Primal and dual functions with a duality gap.Figure 14.6 Primal function with no duality gap.Figure 14.7 Primal function for Example 14.5 with no duality gap.

15 Chapter 15Figure 15.1 Problems with optimal solutions in the interior and on an edge....Figure 15.2 Gradient search.Figure 15.3 Moving to the interior point or the boundary point .