David G Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 2 ppt

... 3 x 1 12 z = –3 z = –3 1 2 z = 2 z = –1 2 1 x 2 Fig. 2. 5 Illustration of extreme point solution 2. 6 Exercises 29 Part 1 2 3 4 5 Casting 2 1331 Finishing 3 2 2 1 1 The profits from the parts are ... min  x i /y i y i > 0   2. 2 Examples of Linear Programming Problems 15 c 1 x 1 +c 2 x 2 +···+c n x n subject to the nutritional constraints a 11 x 1 +a 12 x...

Ngày tải lên: 06/08/2014, 15:20

... chapters. Not only have nonlinear methods improved linear programming, but interior- point methods for linear programming have been extended to provide new approaches to nonlinear programming. This chapter ... show how this merger of linear and nonlinear programming produces elegant and effective methods. These ideas take an especially pleasing form when applied to line...

Ngày tải lên: 06/08/2014, 15:20

... Search 23 0 8.6. The Method of Steepest Descent 23 3 Linear and Nonlinear Programming Third Edition David G. Luenberger Stanford University Yinyu Ye Stanford University 123 PART I LINEAR PROGRAMMING 1.4 ... of nonlinear programming. Linear Programming Linear programming is without doubt the most natural mechanism for formulating a vast array of problems with...

Ngày tải lên: 06/08/2014, 15:20

... problem x 2 x 3 x 4 x 5 b 01−11 3 12 10 2 c T 23 −11−14 Initial tableau—phase II Transforming the last row appropriately we proceed with: 01−11 3 1 2 −10 2 0 22 0 21 First tableau—phase II −1 /20 −1 /21 2 1 /21 −1 /20 1 10 ... decreasing objective, the algorithm must reach a basis satisfying one of the two terminating conditions. Example 1. Maximize 3x 1 +x 2 +3x 3 subject to 2...

Ngày tải lên: 06/08/2014, 15:20

... x 1 + x 2 + 2y 1  4 x 2 + y 1 + y 2  3 2x 1 + x 2  4 x 1 + x 2  2 y 1 + y 2  2 3y 1 +2y 2  5 x 1  0x 2  0y 1  0y 2  0 The decomposition algorithm can be applied by introducing slack ... 8 3 /2 1 0 1 −1 /21 −101−1 12 2001 110 The optimal solution is x 1 =0, x 2 =1, x 3 =2. The corresponding dual program is maximize 4 1 +6 2 subject to 2 1 +  2...

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... REDUCTION OF LINEAR INEQUALITIES Linear programming is in part the study of linear inequalities, and each progressive stage of linear programming theory adds to our understanding of this important fundamental ... restricted primal by pivoting as indicated we obtain a 1 a 2 a 3 ··b 1 121 03 101−1 12 −10− 120 2 −1 /20 3 /2 ··· Now we again calculate the ratios 1 2  3 2...

Ngày tải lên: 06/08/2014, 15:20

... positive, indicating that the current solution is optimal. 3 46895 2 24 553 22 2 321 33 2 4 2 2 2 −1 020 Degeneracy As in all linear programming problems, degeneracy, corresponding to a basic variable ... writing the constraint equations in standard form: x 11 +x 12 +···+x 1n =a 1 x 21 +x 22 +···+x 2n =a 2    x m1 +x m2 +···+x mn =a m x 11 +x 21 x m1 =b 1 x 12...

Ngày tải lên: 06/08/2014, 15:20

... 171 (a) (b) (c) (d) (e) (4, 1) (3, 1) 2 1 1 1 3 1 2 2 2 3 5 4 1 6 (1, 2) (–, ∞) (2, 1) (2, 1)(1, 1) (4, 1) 2 2 3 4 5 6 1 1 1 1 1 1 1 1 3 2 2 (–, ∞) (4, 1) (3, 2) (3, 1) (1, 2) (5, 1) 2 3 3 4 5 6 2 2 2 1 1 1 1 1 1 1 1 1 1 1 (–, ... installing generating equipment, such as (1) nuclear or (2) coal-fired, or by purchasing power from a central energy grid. Associated with t...

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... algorithms for solving nonlinear programming problems are not globally convergent in their purest form and thus occasionally generate sequences that either do not converge at all or converge ... GLOBAL CONVERGENCE OF DESCENT ALGORITHMS A good portion of the remainder of this book is devoted to presentation and analysis of various algorithms designed to solve nonlinear programming...

Ngày tải lên: 06/08/2014, 15:20

... −Ex k+1  Ex k  = 2 g T k g k  2 g T k Qg k  − g T k g k  2 g T k Qg k  g T k Q −1 g k = g T k g k  2 g T k Qg k  g T k Q −1 g k   In order to obtain a bound on the rate of convergence, we ... form x k+1 =x k −  g T k g k g T k Qg k  g k  ( 32) where g k =Qx k −b. 21 8 Chapter 8 Basic Descent Methods N = 2 N = 3 N = 4 N = 5 4 4 5 3 3...

Ngày tải lên: 06/08/2014, 15:20