David G Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 10 potx

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 10 potx

... extension of linear programming. In linear programming, the variables form a vector which is required to be component-wise nonnegative, while in semidefinite programming the variables are compo-nents ...  higher than the minimalobjective cost.Example 2 (Linear Programming) . To see that the problem (SDP) (that is,(56)) generalizes linear programing define C = diagc1c 2 cn, and ... satisfying the first-order conditions for problems when fx and g ix are not generally convexfunctions.Quadratic Programming Let fx = 1 /2 xTQx +cTx and g ix =−xifor i = 1n, and...

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 5 potx

... 73⎤⎦ and finallyP =111⎡⎢⎢⎣1 − 310 −39−301 − 310 0000⎤⎥⎥⎦ (22 )The gradient at the point (2, 2, 1, 0) is g = 2 4 2 −3 and hence we findd =−Pg =111−8 24  −8 0 12. 4 The Gradient ... problemminimize x 2 1+x 2 2+x 2 3+x 2 4−2x1−3x4subject to 2x1+x 2 +x3+4x4=7 (20 )x1+x 2 +2x3+x4=6xi 0i=1 2 3 4Suppose that given the feasible point x = 2 2 1 0 we ... and therefore g 2 =0 is adjoined to the set of working constraints. g 1 = 0∇f T g 2 = 0xFeasible region g 1TFig. 12. 4 Constraint to be dropped11.9 Zero-Order Conditions and Lagrange...

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 13 potx

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David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 1 Part 10 pot

... −Exk+1Exk= 2 g Tk g k 2 g TkQgk− g Tk g k 2 g TkQgk g TkQ−1 g k= g Tk g k 2 g TkQgk g TkQ−1 g kIn order to obtain a bound on the rate of convergence, we ... formxk+1=xk− g Tk g k g TkQgk g k ( 32) where g k=Qxk−b. 21 8 Chapter 8 Basic Descent MethodsN = 2 N = 3N = 4N = 5445333 2 2 2 21111181513 2 31 2 2535 2 83858Fig. ... convergence, and a guarantee of global convergence.8.4 CLOSEDNESS OF LINE SEARCHALGORITHMSSince searching along a line for a minimum point is a component part of most nonlinear programming algorithms,...

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 1 pot

... 2 149690 2 06 023 46 2 17 027 2 2 149693 2 06 023 77 2 1 727 86 2 167983 2 1656418 2 17 427 9 2 173169 2 1657049 2 174583 2 1743 92 2 168440 10 2 174638 2 174397 2 17398111 2 174651 2 1745 82 ... 2 1745 82 2 174048 12 2 174655 2 174643 2 17405413 2 174658 2 174656 2 17460814 2 174659 2 174656 2 17460815 2 174659 2 174658 2 174 622 16 2 174659 2 17465517 2 174659 2 17465618 ... −34198 02 6 −3133619 −3 42 98658 − 324 9978 −3 42 99989 − 329 0408 −343000015 −3396 124 20 −3419 022 25 −3 42 600430 −3 42 83 72 35 −3 42 927 540 −3 42 965045 −3 42 9 825 50 −3 42 990955 −3 42 995160...

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

... hypothesis both g k and Qdkbelong to g 0 Qg0Qk+1 g 0, thefirst by (a) and the second by (b). Thus g k+1∈ g 0 Qg0Qk+1 g 0. Furthermore g k+1 g 0 Qg0Qk g 0 =d0 ... g 0 g 1 g k = g 0 Qg0Qk g 0b) d0 d1dk = g 0 Qg0Qk g 0c) dTkQdi=0 for i  k −1d) k =g Tk g k/dTkQdke) k =g Tk+1 g k+1 /g Tk g k.Proof. ... inSection 10. 4.Scaling and Partial MethodsConvergence of the partial conjugate gradient method, restarted every m +1 steps,will in general be linear. The rate will be determined by the eigenvalue...

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 3 pot

... 10 −451 21 9515 10 −55 25 1115 10 −15 25 1115 10 −14 726 918 10 −6 62 457944 10 −73 323 745 10 −13 323 745 10 −176150890 10 −381 027 00 10 −383 025 393 10 −3 2 973 021 10 −393 025 476 10 −519501 52 10 −3 10 ... Self-scaling1 96.30669 96.30669 96.30669 96.30669 2 1.564971 6994 023 10 −16994 023 10 −169 020 72 10 −1 32 939804 10 2 1 22 5501 10 2 1 22 5501 10 2 3989507 10 −345 810 123 10 −473 0108 8 10 −373 0108 8 10 −3168 426 3 10 −55116 920 5 10 −5 2 636716 ... 97.33665 2 1.58 625 1 1. 621 908 1. 621 908 0.7 024 8 72 32 989875 10 2 8 26 8893 10 −18 26 8893 10 −14090350 10 −34590 8101 10 −443 029 43 10 −143 029 43 10 −11779 424 10 −551194144 10 −544498 52 10 −34449852...

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 4 pps

... problemminimize 2x 2 1+2x1x 2 +x 2 2−10x1−10x 2 subject to x 2 1+x 2 2 53x1+x 2  6The first-order necessary conditions, in addition to the constraints, are4x1+2x 2 10 +2 1x1+3 2 =02x1+2x 2 10 +2 1x 2 + 2 =01 ... yields the equations4x1+2x 2 10 +2 1x1=02x1+2x 2 10 +2 1x 2 =0x 2 1+x 2 2=5which has the solutionx1=1x 2 =2 1=1This yields 3x1+x 2 = 5 and hence the second constraint ... the problemextremize x1+x 2 2+x 2 x3+2x 2 3subject to1 2 x 2 1+x 2 2+x 2 3 =1The first-order necessary conditions are1+ x1=02x 2 +x3+x 2 =0x 2 +4x3+x3=0One solution...

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 6 pps

... −3666061 10 −6645499 10 −3756 423 20 −6656377 20 −3759 123 40 −6658443 50 −3765 128 60 −6659191 100 −3771 625 80 −6659514 20 0 −3778983 100 −6659656 500 −3787989 120 −6659 825 100 0 −37930 12 121 ... returning tothe feasible region from points outside this region. The type of iterative techniqueemployed is a common one in nonlinear programming, including interior-pointmethods of linear programming, ... the projectednegative gradient was computed:minimize x 2 1+x 2 2+x 2 3+x 2 4−2x1−3x4subject to 2x1+x 2 +x3+4x4=7x1+x 2 +2x3+x4=6xi 0i=1 2 3 4We are given the feasible...

David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 7 doc

... 3885653 8 24 3885635 3 15 3885637 3 21 388563c = 20 0⎧⎪⎪⎨⎪⎪⎩1 23 0∗ 23 0 4886073 21 63 4874465 4 20 4874387 2 14 487433c = 20 00⎧⎪⎪⎨⎪⎪⎩1 26 0∗ 26 0 525 23 8345∗135 ... setstrategy. See Gill, Murray, and Wright [G7 ] for a discussion of working sets and active setstrategies. 12. 5 This material is taken from Luenberger [L14]. 12. 6– 12. 7 The reduced gradient method ... study of the global convergence properties of feasibledirection methods was begun by Topkis and Veinott [T8] and by Zangwill [Z2]. 12. 3– 12. 4 The gradient projection method was proposed and developed...

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