In this paper, we present a general algorithm for nonlinear programming which uses a slanting filter criterion for accepting the new iterates. Independently of how these iterates are computed, we prove that all accumu...
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In this paper, we present a general algorithm for nonlinear programming which uses a slanting filter criterion for accepting the new iterates. Independently of how these iterates are computed, we prove that all accumulation points of the sequence generated by the algorithm are feasible. Computing the new iterates by the inexact restoration method, we prove stationarity of all accumulation points of the sequence. (c) 2007 Elsevier Inc. All rights reserved.
In this article, we present generalizations of the cone-preinvexity functions and study a pair of second-order symmetric solutions for multiple objective nonlinear programming problems under these generalizations of t...
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In this article, we present generalizations of the cone-preinvexity functions and study a pair of second-order symmetric solutions for multiple objective nonlinear programming problems under these generalizations of the cone-preinvexity functions. In addition, we establish and prove the theorems of weak duality, strong duality, strict converse duality, and self-duality by assuming the skew-symmetric functions under these generalizations of the cone-preinvexity functions. Finally, we provide four nontrivial numerical examples to demonstrate that the results of the weak and strong duality theorems are true.
Adaptive refinement usually involves refining or enriching a fraction of mesh elements by one level based on a cut-off criterion, requiring several costly intermediate solutions before a mesh that yields an acceptable...
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Adaptive refinement usually involves refining or enriching a fraction of mesh elements by one level based on a cut-off criterion, requiring several costly intermediate solutions before a mesh that yields an acceptable solution is obtained. We avoid this by formulating and solving the mesh design problem as a mathematical program. Our approach simultaneously modifies both mesh size (h) and local polynomial order (p) to yield an "optimal" mesh for a target error or given computational cost with gradients from local convergence rates. Constraints such as the one irregularity rule during mesh refinement are systematically incorporated in this formulation. The design task leads to a mixed integer nonlinear program (MINLP), that is relaxed to an NLP. To reduce the computations for the NLP, we employ simplified analytical gradients derived from initial mesh calculations. Finally, we apply our method to three model problems showing that complex hp-adaptive grids can be obtained directly from a uniform coarse grid. A commercial optimization software, MINOS [B.A. Murtagh, M.A. Saunders, MINOS 5.4 User's Guide, Technical Report SOL 83-20R, Stanford University, Stanford, 1987, Revised February 1995], was used as the NLP optimizer. (C) 2001 Elsevier Science B.V. All rights reserved.
We analyze the rate of convergence of the proximal method of multipliers for non-convex nonlinear programming problems. First, we prove, under the strict complementarity condition, that the rate of convergence of the ...
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We analyze the rate of convergence of the proximal method of multipliers for non-convex nonlinear programming problems. First, we prove, under the strict complementarity condition, that the rate of convergence of the proximal method of multipliers is linear and the ratio constant is proportional to 1/c when the ratio is small enough, which implies that the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to . Second, we prove that, without strict complementarity condition, the rate of convergence of the proximal method of multipliers is proportional to 1/c when c exceeds a threshold.
As noted by Wachter and Biegler (Ref. 1), a number of interior-point methods for nonlinear programming based on line-search strategy may generate a sequence converging to an infeasible point. We show that, by adopting...
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As noted by Wachter and Biegler (Ref. 1), a number of interior-point methods for nonlinear programming based on line-search strategy may generate a sequence converging to an infeasible point. We show that, by adopting a suitable merit function, a modified primal-dual equation, and a proper line-search procedure, a class of interior-point methods of line-search type will generate a sequence such that either all the limit points of the sequence are KKT points, or one of the limit points is a Fritz John point, or one of the limit points is an infeasible point that is a stationary point minimizing a function measuring the extent of violation to the constraint system. The analysis does not depend on the regularity assumptions on the problem. Instead, it uses a set of satisfiable conditions on the algorithm implementation to derive the desired convergence property.
Water allocation under limited water supplies is becoming more important as water becomes scarcer. Optimization models are frequently used to provide decision support to enhance water allocation under limited water su...
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Water allocation under limited water supplies is becoming more important as water becomes scarcer. Optimization models are frequently used to provide decision support to enhance water allocation under limited water supplies. Correct modelling of the underlying soil-moisture balance calculations at the field scale, which governs optimal allocation of water is a necessity for decision-making. Research shows that the mathematical programming formulation of soil-moisture balance calculations presented by Ghahraman and Sepaskhah (2004) may malfunction under limited water supplies. A new model formulation is presented in this research that explicitly models deep percolation and evapotranspiration as a function of soil-moisture content. The new formulation also allows for the explicit modelling of inefficiencies resulting from nonuniform irrigation. Modelling inefficiencies are key to the evaluation of the economic profitability of deficit irrigation. Ignoring increasing efficiencies resulting from deficit irrigation may render deficit irrigation unprofitable. The results show that ignoring increasing efficiencies may overestimate the impact of deficit irrigation on maize yields by a maximum of 2.2 tons per hectare.
An important challenge for most chemical companies is, to simultaneously consider inventory optimization and supply chain network design under demand uncertainty. This leads to a problem that requires integrating a st...
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An important challenge for most chemical companies is, to simultaneously consider inventory optimization and supply chain network design under demand uncertainty. This leads to a problem that requires integrating a stochastic inventory model with the supply chain network design model. This problem can be formulated as a large-scale combinatorial optimization model that includes nonlinear terms. Since these models are very difficult to solve. they require exploiting their properties and developing special solution techniques to reduce the computational effort. In this work, we analyze the properties of the basic model and develop solution techniques for a joint Supply chain network design and inventory management model for a given product. The model is formulated as a nonlinear integer programming problem. By reformulating it as a mixed-integer nonlinear programming (MINLP) problem and using an associated convex relaxation model for initialization. we first propose a heuristic method to quickly obtain good-quality solutions. Further. a decomposition algorithm based on Lagrangean relaxation is developed for obtaining global or near-global optimal solutions. Extensive computational examples with up to 150 distribution centers and 150 retailers are presented to illustrate the performance of the algorithms and to compare them with the full-space solution.
A novel idea is proposed for solving optimization problems with equality constraints and bounds on the variables. In the spirit of sequential quadratic programming and sequential linearly-constrained programming, the ...
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A novel idea is proposed for solving optimization problems with equality constraints and bounds on the variables. In the spirit of sequential quadratic programming and sequential linearly-constrained programming, the new proposed approach approximately solves, at each iteration, an equality-constrained optimization problem. The bound constraints are handled in outer iterations by means of an augmented Lagrangian scheme. Global convergence of the method follows from well-established nonlinear programming theories. Numerical experiments are presented.
Line search methods are proposed for nonlinear programming using Fletcher and Leyffer's filter method [ Math. Program., 91 ( 2002), pp. 239 - 269], which replaces the traditional merit function. Their global conve...
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Line search methods are proposed for nonlinear programming using Fletcher and Leyffer's filter method [ Math. Program., 91 ( 2002), pp. 239 - 269], which replaces the traditional merit function. Their global convergence properties are analyzed. The presented framework is applied to active set sequential quadratic programming (SQP) and barrier interior point algorithms. Under mild assumptions it is shown that every limit point of the sequence of iterates generated by the algorithm is feasible, and that there exists at least one limit point that is a stationary point for the problem under consideration. A new alternative filter approach employing the Lagrangian function instead of the objective function with identical global convergence properties is briefly discussed.
In this paper, we explore the verification problem of outsourcing constrained nonlinear programming (NLP) when it is required to be solved by particle swarm optimization (PSO) algorithm, i.e., making sure that the clo...
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In this paper, we explore the verification problem of outsourcing constrained nonlinear programming (NLP) when it is required to be solved by particle swarm optimization (PSO) algorithm, i.e., making sure that the cloud runs PSO algorithm faithfully and returns an acceptable solution. An efficient verification scheme without any cryptographic tool is proposed. The proposed scheme involves approximate KKT conditions with the epsilon-KKT point in verifying the optimality of the result returned by PSO algorithm. Extensive experiments on PSO benchmarks and NLP test problems demonstrate that our proposed scheme is effective and efficient at verifying the cloud's honesty.
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