By R. Tyrrell Rockafellar
A rigorous and complete remedy of community move idea and monotropic optimization by way of one of many world's most famed utilized mathematicians.
This vintage textbook, first released by way of J. Wiley & Sons, Inc., in 1984, covers generally the duality thought and the algorithms of linear and nonlinear community optimization optimization, and their major extensions to monotropic programming (separable convex restricted optimization difficulties, together with linear programs).
Monotropic programming difficulties are characterised by way of a wealthy interaction among combinatorial constitution and convexity houses. Rockafellar develops, for the 1st time, algorithms and a remarkably entire duality conception for those difficulties.
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Extra info for Network flows and monotropic optimization
72). This case is studied in Sect. 3 by combining the above mentioned results for an H 1 observation with the statespace regularization technique introduced in State-space Regularization of Sect. 4. Chapter 2 Computing Derivatives We address in this chapter a practical aspect of the numerical resolution of NLS problems, namely the computation of the gradient of the objective function or the Jacobian of the forward map, after discretization has occurred. This calculation has to be computed both accurately, so that the optimization algorithm has a chance to work properly, and eﬃciently, in order to keep computation time as low as possible.
4 Example 2: 1D Elliptic Parameter Estimation Problem Our second example is inﬁnite dimensional: it consists in estimating the coeﬃcient a as a function of the space variable ξ in the one-dimensional elliptic 22 CHAPTER 1. 38) ⎩ J is a ﬁnite set of source indexes. 37) with Dirichlet boundary conditions: u(0) = 0, u(1) = 0. 38) models, for example, the temperature u in a one-dimensional slab at thermal equilibrium, heated by point sources of amplitude gi at locations xi , and whose temperature is maintained equal to zero at each end, in which case a is the thermal diﬀusion coeﬃcient.
Step 1: State-Space Decomposition We use here the decomposition suggested in Sect. 1: y = (e, f, S1 , S2 , . . 29) M = [0 . . 30) 42 CHAPTER 2. COMPUTING DERIVATIVES which has to be complemented by the state equation e(x, y) = 0. 2) in the form of a sequence of equations. 31) (N1 ≥ 0) N1 + q − T1 = 0 ⎪ 2 2 ⎪ + q − T = 0 (N ≥ 0) N ⎪ 2 2 2 ⎪ ⎪ 2 ⎪ D − eq = 0 ⎪ ⎪ ⎪ ⎪ A − eρ + D = 0 ⎪ ⎪ ⎪ ⎪ K −D+A=0 ⎪ ⎪ ⎪ ⎪ B−1+K = 0 ⎪ ⎪ ⎪ ⎪ ⎪ C −1−K =0 ⎪ ⎪ ⎪ ⎪ P − M1 (B 2 N1 + f N2 ) − 4eDM1 M2 N1 N2 = 0 ⎪ ⎪ ⎪ ⎪ Q − M2 (C 2 N2 + f N1 ) − 4q 2 A2 = 0 ⎪ ⎩ (P + Q)R − (P − Q) = 0.
Network flows and monotropic optimization by R. Tyrrell Rockafellar