By Jean-Baptiste Hiriart-Urruty, Claude Lemarechal

From the stories: "The account is kind of particular and is written in a way that would attract analysts and numerical practitioners alike...they include every thing from rigorous proofs to tables of numerical calculations.... one of many robust positive factors of those books...that they're designed no longer for the specialist, yet in the event you whish to benefit the subject material ranging from very little background...there are a number of examples, and counter-examples, to again up the theory...To my wisdom, no different authors have given any such transparent geometric account of convex analysis." "This leading edge textual content is easily written, copiously illustrated, and obtainable to a large audience"

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**Extra info for Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods**

**Example text**

XII. II)isa reasonable task, which can be performed in 0 (m) 0 (n 2) operations. Furthermore, the resulting Sk is an integer vector. This example will be used with several datasets corresponding to several values ofn, and referred to as Tspn. 0 = Take first TSP442, having thus n = 442 variables; the minimal value of f turns out to be -50505. 5. With this larger test-problem, Fig. 4 displays an evolution similar to that of TR48; but at least the cheap variant is no longer "non-convergent": the two curves can be drawn on the same picture.

1) with d Replace k by k 31 = dk to obtain sk+1 E S such that (Sk+I' dk) = as(dk) . + I and loop to Step 1. o The black box involved in this form of algorithm is, somehow, an "intelligent (VI)". Its input is no longer x but rather the couple (x, d); its output is no longer an arbitrary s(x) E S = af(x), but a somewhat particular Sd(X), lying in the face of S exposed by d. The initialization is artificial: with d = 0, so(x) is now an arbitrary subgradient. The next idea that comes to mind is: how about computing not only a separating hyperplane, but a "best" one, separating S as much as possible from O?

Unfortunately, this is blatantly contradicted empirically. Some numerical experiments will give an idea of what is observed in practice. 6 (TR48) Given a symmetric n x n matrix A, and two vectors d and s in Rn , consider the piecewise affine objective function n f(x) := I)SiXi + di max {ail - XJ, ai2 - X2, ... , ain - xn}] . 4 is performed on the mathematical formulation of this problem). 10) is not a particularly good one for obtaining effective solutions of transportation problems; but it is an excellent academic example to illustrate some of our points.