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On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian
 Journal of Global Optimization
, 2006
"... We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condit ..."
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Cited by 4 (2 self)
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We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the stepsize parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. Key words: Nonconvex programming; nonsmooth optimization; augmented Lagrangian; sharp Lagrangian; subgradient optimization.
Algorithms for the Quasiconvex Feasibility Problem Yair Censor
, 2004
"... We study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x ∗ ∈ Rn that satisfies the inequalities f1(x∗) ≤ 0, f2(x∗) ≤ 0,..., fm(x∗) ≤ 0, where all functions are continuous and quasiconvex. We consider the consistent case when the so ..."
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Cited by 1 (1 self)
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We study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x ∗ ∈ Rn that satisfies the inequalities f1(x∗) ≤ 0, f2(x∗) ≤ 0,..., fm(x∗) ≤ 0, where all functions are continuous and quasiconvex. We consider the consistent case when the solution set is nonempty. Since the FenchelMoreau subdifferential might be empty we look at different notions of the subdifferential and determine their suitability for our problem. We also determine conditions on the functions, that are needed for convergence of our algorithms. The quasiconvex functions on the lefthand side of the inequalities need not be differentiable but have to satisfy a Lipschitz or a Hölder condition. 1
COST MINIMIZATION IN MULTI−COMMODITY, MULTI−MODE Approved by: GENERALIZED NETWORKS WITH TIME WINDOWS
, 2005
"... Chair of Advisory Committee: Dr. Alberto GarciaDiaz The purpose of this research is to develop a heuristic algorithm to minimize total costs in multicommodity, multimode generalized networks with time windows problems. The proposed mathematical model incorporates features of the congestion of veh ..."
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Chair of Advisory Committee: Dr. Alberto GarciaDiaz The purpose of this research is to develop a heuristic algorithm to minimize total costs in multicommodity, multimode generalized networks with time windows problems. The proposed mathematical model incorporates features of the congestion of vehicle flows and time restriction of delivering commodities. The heuristic algorithm, HA, has two phases. Phase 1 provides lower and upper bounds based on Lagrangian relaxations with subgradient methods. Phase 2 applies two methods, early due date with overduedate costs and total transportation costs, to search for an improved upper bound. Two application networks are used to test HA for small and mediumscale problems. A different number of commodities and various lengths of planning time periods are generated. Results show that HA can provide good feasible solutions within the reasonable range of optimal solutions. If optimal solutions are unknown, the average gap between lower and upper bounds is 0.0239. Minimal and maximal gaps are 0.0007 and 0.3330. If optimal solutions are known, the maximal gap between upper bounds and optimal solutions is less than 10 % ranges of optimal solutions.