Nonlinear programming

... LECTURE SLIDES ON NONLINEAR PROGRAMMING BASED ON LECTURES GIVEN AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASS DIMITRI P BERTSEKAS These lecture slides are based on the book: Nonlinear ... Quadratic Approximation of f at x1 Slow convergence of steepest descent Fast convergence of Newton’s method w/ αk = Given xk , the...

Ngày tải lên: 26/10/2013, 17:15

... Mathematics and its Applications MIXED INTEGER NONLINEAR PROGRAMMING contains expository and research papers based on a highly successful IMA Hot Topics Workshop Mixed- Integer Nonlinear Optimization: ... of n-fold integer programming (S Onn) is an overview of the theory of n-fold integer programming, which enables the polynomial-time solution of fundamental linear and non...

Ngày tải lên: 30/05/2014, 00:33

... assumptions, and preliminary results In Section 3, we obtain an approximate KKT optimality condition of generalized augmented Lagrangian problem and prove that the approximate stationary points of the generalized ... unconstrained programming problem formulated by using a generalized augmented Lagrangian function As an application, an approximate KKT optimality c...

Ngày tải lên: 22/06/2014, 18:20

... divided into three major parts: Linear Programming, Unconstrained Problems, and Constrained Problems The last two parts together comprise the subject of nonlinear programming Linear Programming Linear ... Models and Principles ∗ A list of the early publications in the series is at the end of the book∗ Linear and Nonlinear Programming Third Edition David G Lue...

Ngày tải lên: 27/06/2014, 17:20

... Mathematics and its Applications MIXED INTEGER NONLINEAR PROGRAMMING contains expository and research papers based on a highly successful IMA Hot Topics Workshop Mixed- Integer Nonlinear Optimization: ... of n-fold integer programming (S Onn) is an overview of the theory of n-fold integer programming, which enables the polynomial-time solution of fundamental linear and non...

Ngày tải lên: 01/08/2014, 16:47

... 5 .1 ∗ 5.2 ∗ 5.3 5.4 5.5 5.6 5.7 5.8 5.9 93 98 10 3 11 1 PART II Chapter 11 2 11 4 11 5 11 8 12 1 12 6 13 4 13 9 14 0 Transportation and Network Flow Problems 14 5 6 .1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6 .10 ... Subspace 11 .7 Sensitivity 11 .8 Inequality Constraints 11 .9 Zero-Order Conditions and Lagrange Multipliers 11 .10 Summary 11 .11 Exercises Chapter 12 Primal...

Ngày tải lên: 06/08/2014, 15:20

... x1 + x2 − x3 = 2x1 − 3x2 + x3 = −x1 + 2x2 − x3 = 1 To obtain an original basis, we form the augmented tableau e1 0 e2 e3 0 a1 a2 a3 b 1 1 −3 1 1 1 and replace e1 by a1 e2 by a2 , and e3 ... 2. 2 Examples of Linear Programming Problems 15 c x + c2 x2 + · · · + c n x n subject to the nutritional constraints a 11 x1 + a 12 x2 + · · · + a1n xn a 21 x1 + a 22 x2 + · · · + a2n...

Ngày tải lên: 06/08/2014, 15:20

... Method x2 1 1 x3 1 −2 x4 1 x5 0 x6 1 x7 b −2 1 Second tableau—phase I 1 1 1 0 1 Final tableau—phase I Now we go back to the equivalent reduced problem cT x3 x2 x4 1 1 1 x5 1 b 14 Initial ... 1 1 xB 0 1 Now T = 1 3 B 1 = −2 and r1 = −7 r4 = 3 r5 = ∗ 3. 8 The Simplex Method and LU Decomposition 59 We select a1 to enter the basis We have the tableau B 1 Variable −2...

Ngày tải lên: 06/08/2014, 15:20

... usual, we obtain the new tableau and new follows B 1 Variable Value 0 2 s1 3 T 1 1/2 1/ 2 0 0 1 1 1/ 2 1/ 2 1/ 2 0 s1 1/ 2 1/ 2 −2 −2 1 1 1/ 2 1/ 2 = 4 −2 −7 B 1 = 1 −2 −5 as 70 Chapter The Simplex ... explanation 1 2 4 1 0 3/2 1 2 −3 0 0 0 1/ 2 1/ 2 1 1 2 1/ 2 1 10 1 The optimal solution is x1 = 0, x2 = 1, x3 = The corresponding dual program is maximize subject to...

Ngày tải lên: 06/08/2014, 15:20

... /y2j and select the minimum positive ratio This yields the pivot indicated Continuing, the remaining tableaus are 0 − 1 1 0 5/ 2 1/ 2 7/2 1/ 2 1/ 2 3/2 −2 Second tableau 5/ 2 1 1/2 −2 1 1 11 0 ... problem, and the corresponding dual solution is ˆ = − r ii) Show that this scheme fully allocates H 12 Solve the linear inequalities −2x1 + 2x2 2x1 − x2 − 4x2 15 x1 − 12 x2 12 x1 +...

Ngày tải lên: 06/08/2014, 15:20

... Ek +1 is defined to be the minimal-volume ellipsoid containing 1/ 2 Ek It is constructed as follows Define = m +1 = yk m2 m2 − 1/ 2 E Fig 5 .1 A half-ellipsoid =2 ∗ 5.3 The Ellipsoid Method 11 7 Then ... m +1 The reduction in volume is the product of the square roots m2 1 of these, giving the equality in the theorem Then using + x p exp , we have m2 m2 − m 1 /2 m = 1+ m +1 m 1 < e...

Ngày tải lên: 06/08/2014, 15:20

... constraint equations in standard form: x 11 + x12 + · · · + x1n = a1 = a2 x 21 + x22 + · · · + x2n xm1 + xm2 + · · · + xmn = am + x 21 x 11 xm1 + x22 x12 x1n + xm2 + x2n = b1 = b2 (3) + xmn = bn The ... Goldfarb and Xiao [G 11] , Gonzaga and Todd [G14], Todd [T4], Tun el [T10], Tutuncu [T 11] , and others The homogeneous and self-dual embedding method can be found in Ye et al [Y...

Ngày tải lên: 06/08/2014, 15:20

... (1, 2) (3, 1) (b) (4, 1) (3, 1) 2 (–, ∞) 1 1 1 3 (1, 1) (2, 1) (c) (d) (–, ∞) (5, 1) 1 2 (e) Fig 6.6 Example of maximal flow problem (5, 1) 17 1 17 2 Chapter Transportation and Network Flow Problems ... source, and node is the sink The original network with capacities indicated on the 6 .8 Maximal Flow (1, 1) (2, 1) 2 (–, ∞) 1 (4, 1) (1, 2) (2, 1) (a) (4, 1) (3, 2)...

Ngày tải lên: 06/08/2014, 15:20

... at x0 = 10 0, each of the sequences below might be generated from iterative application of this algorithm 10 0 50 25 12 −6 −2 1/ 2 10 0 −40 20 −5 −2 1/ 4 1/ 8 10 0 10 1 1 /16 1/ 100 1/ 1000 1/ 10 000 The ... , alternatively y = x1 or y = x2 , we have Setting x = x1 + − x2 and f x1 f x1 + − (10 ) f x2 Multiplying (10 ) by f x + f x x1 − x f x + f x x2 − x (11 ) and (11 ) by (1...

Ngày tải lên: 06/08/2014, 15:20

... write (13 ) as xk +1 − x∗ = xk − x∗ xk 1 − x∗ g xk 1 xk x∗ g xk 1 xk Now, by the mean value theorem with remainder, we have (see Exercise 2) g xk 1 xk = g (14 ) k and g xk 1 xk x∗ = g where k and tively ... Q 1 x n i =1 n i =1 xi2 i xi n i =1 xi2 / i which can be written as xT x 1/ n = n i =1 i i ≡ xT Qx xT Q 1 x i =1 i / i where i = xi2 / n xi2 We have converted the express...

Ngày tải lên: 06/08/2014, 15:20