Computational Optimization: A Tribute to Olvi Mangasarian by Jong-Shi Pang (auth.), Jong-Shi Pang (eds.)

By Jong-Shi Pang (auth.), Jong-Shi Pang (eds.)

Computational Optimization: A Tribute to Olvi Mangasarian serves as an exceptional reference, supplying perception into essentially the most not easy learn concerns within the box.
This choice of papers covers a large spectrum of computational optimization subject matters, representing a mix of general nonlinear programming subject matters and such novel paradigms as semidefinite programming and complementarity-constrained nonlinear courses. Many new effects are offered in those papers that are certain to motivate extra study and generate new avenues for functions. an off-the-cuff categorization of the papers contains:

  • Algorithmic advances for targeted sessions of restricted optimization difficulties
  • Analysis of linear and nonlinear courses
  • Algorithmic advances
  • B- desk bound issues of mathematical courses with equilibrium constraints
  • Applications of optimization
  • Some mathematical subject matters
  • Systems of nonlinear equations.

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Additional info for Computational Optimization: A Tribute to Olvi Mangasarian Volume I

Example text

R q* = supq(/l), (3) flC':O where q : ))tr q(/l) = f--+ [-00, +00) is the dual function given by inf{f(x) XEX + /l'g(x)}. (4) Throughout the paper, we assume the following: Assumption 1. , g(XF) :::: O. , gj(xd > o for at least one j. Furthermore, its cost f(x,) is strictly sma lIe r than the cost f (x F) of XF. We note that by weak duality, we have q* :::: f(XF). We will show that the value f(x,) can be used to improve this upper bound. In particular, we prove the following result in Section 3: Proposition 1.

V: (y, u):s (y. x). Vu E C} ifxEC otherwise. n int C =j:. 10) that V(x, g) E G(i), p ( l(k-l)2) k k k k k-l i x Adx-x ,g-e}::: 8(li(X )-li(x» 2li(x )-li(x ) - Ii (xk) . 11) (x, g) = (x*, 0) with x* E S. Applying Lemma 2 componentwise with s =li(x k- 1), t =li(x k ) and u =li(x*), and summing over i = 1, ... , p, we then obtain Ak(X* - xk, -i} ::: ~(IIA(X* - x k )1I2 - IIA(x* - x k- 1) 112) + ~ IIA(x k _ x k- 1)1I2. 4), (recall that Ak ::: A > 0) it follows from Lemma 1 that the sequence {II A (x* - xk) II} converges.

In all the above papers, convergence was proved under restrictive assumptions on the problem data. In fact the challenge remains to generate an interior proximal method which is globally convergent to a solution of (VI), under the only assumption that the set of solutions of (VI) is nonempty. Indeed, until now this objective was not attained. In [16] only ergodic convergence is proved and in [1, 4-6], it is assumed that the map T is paramonotone that is: (x, y), (x', y') E G (T) and (x - x', y - y') = 0 collectively imply that (x, y') E G(T).

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