A generalized approach is taken to linear and quadraticprogramming in which dual as well as primal variables may be subjected to bounds, and constraints may be represented through penalties. Corresponding problem mod...
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A generalized approach is taken to linear and quadraticprogramming in which dual as well as primal variables may be subjected to bounds, and constraints may be represented through penalties. Corresponding problem models in optimal control related to continuous-time programming are then set up and theorems on duality and the existence of solutions are derived. Optimality conditions are obtained in the form of a global saddle point property which decomposes into an instantaneous saddle point condition on the primal and dual control vectors at each time, along with an endpoint condition.
Mehrotra and Ozevin [SIAM J. Optim., 19 (2009), pp. 1846-1880] computationally found that a weighted primal barrier decomposition algorithm significantly outperforms the equally weighted barrier decomposition proposed...
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Mehrotra and Ozevin [SIAM J. Optim., 19 (2009), pp. 1846-1880] computationally found that a weighted primal barrier decomposition algorithm significantly outperforms the equally weighted barrier decomposition proposed and analyzed in [G. Zhao, Math. Program., 90 (2001), pp. 507-536;S. Mehrotra and M. G. Ozevin, Oper. Res., 57 (2009), pp. 964-974;S. Mehrotra and M. G. Ozevin, SIAM J. Optim., 18 (2007), pp. 206-222]. Here we consider a weighted barrier that allows us to analyze iteration complexity of algorithms in all of the aforementioned publications in a unified framework. In particular, we prove self-concordance parameter values for the weighted barrier and using these values give a worst-case iteration complexity bound for the weighted decomposition algorithm.
Zhao showed that the log barrier associated with the recourse function of two-stage stochastic linear programs behaves as a strongly self-concordant barrier and forms a self-concordant family on the first-stage soluti...
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Zhao showed that the log barrier associated with the recourse function of two-stage stochastic linear programs behaves as a strongly self-concordant barrier and forms a self-concordant family on the first-stage solutions. In this paper, we show that the recourse function is also strongly self-concordant and forms a self-concordant family for the two-stage stochastic convex quadratic programs with recourse. This allows us to develop Bender's decomposition based linearly convergent interior point algorithms. An analysis of such an algorithm is given in this paper.
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