A line search method is proposed for nonlinear programming using Fletcher and Leyffer's filter method, which replaces the traditional merit function. A simple modi. cation of the method proposed in a companion pap...
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A line search method is proposed for nonlinear programming using Fletcher and Leyffer's filter method, which replaces the traditional merit function. A simple modi. cation of the method proposed in a companion paper [SIAM J. Optim., 16 ( 2005), pp. 1 - 31] introducing second order correction steps is presented. It is shown that the proposed method does not suffer from the Maratos effect, so that fast local convergence to second order sufficient local solutions is achieved.
Recently, Fletcher and Leyffer proposed using filter methods instead of a merit function to control steplengths in a sequential quadratic programming algorithm. In this paper, we analyze possible ways to implement a f...
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Recently, Fletcher and Leyffer proposed using filter methods instead of a merit function to control steplengths in a sequential quadratic programming algorithm. In this paper, we analyze possible ways to implement a filter-based approach in an interior-point algorithm. Extensive numerical testing shows that such an approach is more efficient than using a merit function alone.
This article considers the problem of designing a continuous-time dynamical system that solves a constrained nonlinear optimization problem and makes the feasible set forward invariant and asymptotically stable. The i...
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This article considers the problem of designing a continuous-time dynamical system that solves a constrained nonlinear optimization problem and makes the feasible set forward invariant and asymptotically stable. The invariance of the feasible set makes the dynamics anytime, when viewed as an algorithm, meaning it returns a feasible solution regardless of when it is terminated. Our approach augments the gradient flow of the objective function with inputs defined by the constraint functions, treats the feasible set as a safe set, and synthesizes a safe feedback controller using techniques from the theory of control barrier functions. The resulting closed-loop system, termed safe gradient flow, can be viewed as a primal-dual flow, where the state corresponds to the primal variables and the inputs correspond to the dual ones. We provide a detailed suite of conditions based on constraint qualification under which (both isolated and nonisolated) local minimizers are asymptotically stable with respect to the feasible set and the whole state space. Comparisons with other continuous-time methods for optimization in a simple example illustrate the advantages of the safe gradient flow.
Optimality conditions for nonlinear problems with equality and inequality constraints are considered. In the case when no constraint qualification (or regularity) is assumed, the Lagrange multiplier corresponding to t...
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Optimality conditions for nonlinear problems with equality and inequality constraints are considered. In the case when no constraint qualification (or regularity) is assumed, the Lagrange multiplier corresponding to the objective function can vanish in first order necessary optimality conditions given by Fritz John and the corresponding extremum is called abnormal. In the paper we consider second order sufficient optimality conditions that guarantee the rigidity of abnormal extrema (i.e. their isolatedness in the admissible sets). (C) 2006 Elsevier B.V. All rights reserved.
In this paper we present a chaos-based evolutionary algorithm (EA) for solving nonlinear programming problems named chaotic genetic algorithm (CGA). CGA integrates genetic algorithm (GA) and chaotic local search (CLS)...
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In this paper we present a chaos-based evolutionary algorithm (EA) for solving nonlinear programming problems named chaotic genetic algorithm (CGA). CGA integrates genetic algorithm (GA) and chaotic local search (CLS) strategy to accelerate the optimum seeking operation and to speed the convergence to the global solution. The integration of global search represented in genetic algorithm and CLS procedures should offer the advantages of both optimization methods while offsetting their disadvantages. By this way, it is intended to enhance the global convergence and to prevent to stick on a local solution. The inherent characteristics of chaos can enhance optimization algorithms by enabling it to escape from local solutions and increase the convergence to reach to the global solution. Twelve chaotic maps have been analyzed in the proposed approach. The simulation results using the set of CEC'2005 show that the application of chaotic mapping may be an effective strategy to improve the performances of EAs. (C) 2016 Elsevier Ltd. All rights reserved.
In this paper, we construct a new spline smoothing homotopy method for solving general nonlinear programming problems with a large number of complicated constraints. We transform the equality constraints into the ineq...
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In this paper, we construct a new spline smoothing homotopy method for solving general nonlinear programming problems with a large number of complicated constraints. We transform the equality constraints into the inequality constraints by introducing two parameters. Subsequently, we use smooth spline functions to approximate the inequality constraints. The smooth spline functions involve only few inequality constraints. In other words, the method introduces an active set technique. Under some weaker conditions, we obtain the global convergence of the new spline smoothing homotopy method. We perform numerical tests to compare the new method to other methods, and the numerical results show that the new spline smoothing homotopy method is highly efficient.
A very surprising result is derived in this paper, that there exists a family of LP duals for general NLP problems. A general dual problem is first derived from implied constraints via a simple bounding technique. It ...
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A very surprising result is derived in this paper, that there exists a family of LP duals for general NLP problems. A general dual problem is first derived from implied constraints via a simple bounding technique. It is shown that the Lagrangian dual is a special case of this general dual and that other special cases turn out to be LP problems. The LP duals provide a very powerful computational device but are derived using fairly strict conditions. Hence, they can often be infeasible even if the primal NLP problem is feasible and bounded. Many directions for relaxing these conditions are outlined for future research. A concept of local duality is also introduced for the first time akin to the concept of local optimality. (C) 1999 Elsevier Science B.V. All rights reserved.
We propose two line search primal-dual interior-point methods for nonlinear programming that approximately solve a sequence of equality constrained barrier subproblems. To solve each subproblem, our methods apply a mo...
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We propose two line search primal-dual interior-point methods for nonlinear programming that approximately solve a sequence of equality constrained barrier subproblems. To solve each subproblem, our methods apply a modified Newton method and use an tau(2)-exact penalty function to attain feasibility. Our methods have strong global convergence properties under standard assumptions. Specifically, if the penalty parameter remains bounded, any limit point of the iterate sequence is either a Karush-Kuhn-Tucker (KKT) point of the barrier subproblem, or a Fritz-John (FJ) point of the original problem that fails to satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ);if the penalty parameter tends to infinity, there is a limit point that is either an infeasible FJ point of the inequality constrained feasibility problem (an infeasible stationary point of the infeasibility measure if slack variables are added) or a FJ point of the original problem at which the MFCQ fails to hold. Numerical results are given that illustrate these outcomes.
The finite-dimensional McCormick second-order sufficiency theory for nonlinear programming problems with a finite number of constraints is now a classical part of the optimization literature. It was introduced by McCo...
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The finite-dimensional McCormick second-order sufficiency theory for nonlinear programming problems with a finite number of constraints is now a classical part of the optimization literature. It was introduced by McCormick in 1967 and an improved version was given by Fiacco and McCormick in their 1968 award-winning book. Later it was learned that in 1953 Pennisi had presented exactly the same theory. Many authors, most notably Maurer and Zowe in a widely cited paper in 1978, argue that the Pennisi-McCormick theory cannot be extended to infinite dimensions without adding further assumptions, by producing a counterexample. They then extend the theory to infinite dimensions, allowing for an infinite number of constraints, by strengthening the sufficient conditions required. In the current paper we use a fundamental principle for second-order sufficiency to extend the Pennisi-McCormick second-order theory as stated in R-n to infinite-dimensional normed vector spaces, without strengthening the conditions. The Maurer and Zowe infinite-dimensional counterexample carried an infinite number of constraints. Hence they seemed to be unaware that the extension of the Pennisi-McCormick theory to infinite dimensions was possible provided the original feature of a finite number of constraints was maintained.
The ant colony optimization algorithm (ACOA) is hybridized with nonlinear programming (NLP) for the optimal design of sewer networks. The resulting problem is a highly constrained mixed integer nonlinear problem (MINL...
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The ant colony optimization algorithm (ACOA) is hybridized with nonlinear programming (NLP) for the optimal design of sewer networks. The resulting problem is a highly constrained mixed integer nonlinear problem (MINLP) presenting a challenge even to the modern heuristic search methods. In the proposed hybrid method, The ACOA is used to determine pipe diameters while the NLP is used to determine the pipe slopes of the network by proposing two different formulations. In the first formulation, named ACOA-NLP1 a penalty method is used to satisfy the problem constraints while in the second one, named ACOA-NLP2, the velocity and flow depth constraints are expressed in terms of the slope constraints which are easily satisfied as box constraint of the NLP solver leading to a considerable reduction of the search space size. In addition, the assumption of minimum cover depth at the network inlets is used to calculate the nodal cover depths and the pump and drop heights at the network nodes, if required, leading to a complete solution. The total cost of the constructed solution is used as the objective function of the ACOA, guiding the ant toward minimum cost solutions. Proposed hybrid methods are used to solve three test examples, and the results are presented and compared with those produced by a conventional application of ACOA. The results indicate the effectiveness and efficiency of the proposed formulations and in particular the ACOA-NLP2 to optimally solve the sewer network design optimization problems. (C) 2018 Water Environment Federation
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