This report illustrates, by means of numerical examples, the behavior of the constrained minimization algorithm REQP in situations where the active constraint normals are not linearly independent. The examples are int...
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This report illustrates, by means of numerical examples, the behavior of the constrained minimization algorithm REQP in situations where the active constraint normals are not linearly independent. The examples are intended to demonstrate that the presence of the penalty parameter in the equations for calculating the Lagrange multiplier estimates enables a useful search direction to be computed. This is shown to be true, whether the dependence among the constraint normals occurs at the solution or in some other region.
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.
We present a class of trust region algorithms without using a penalty function or a filter for nonlinear inequality constrained optimization and analyze their global and local convergence. In each iteration, the algor...
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We present a class of trust region algorithms without using a penalty function or a filter for nonlinear inequality constrained optimization and analyze their global and local convergence. In each iteration, the algorithms reduce the value of objective function or the. measure of constraints violation according to the relationship between optimality and feasibility. A sequence of steps focused on improving optimality is referred to as an f-loop, while some restoration phase focuses on improving feasibility and is called an h-loop. In an f-loop, the algorithms compute trial step by solving a classic QP subproblem rather than using composite-step strategy. Global convergence is ensured by requiring the constraints violation of each iteration not to exceed an progressively tighter bound on constraints violation. By using a second order correction strategy based on active set identification technique. Marato's effect is avoided and fast local convergence is shown. The preliminary numerical results are encouraging. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.
Many practical problems often lead to large nonconvex nonlinear programming problems that have many equality constraints. The global optimization algorithms of these problems have received much attention over the last...
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Many practical problems often lead to large nonconvex nonlinear programming problems that have many equality constraints. The global optimization algorithms of these problems have received much attention over the last few years. Generally, stochastic algorithms are suitable for these problems, but not efficient when there are too many equality constraints. Therefore, a global optimization algorithm for solving these problems is proposed in this paper. The new algorithm, based on a feasible set strategy, uses a stochastic algorithm and a deterministic local algorithm. The convergence of the algorithm is analyzed. This algorithm is applied to practical problem, and the numerical results illustrate the accuracy and efficiency of the algorithm. (C) 2003 Elsevier Inc. All rights reserved.
A two-step topology-finding method based on mixed integer programming and nonlinear programming is proposed for tensegrity structures. In the first step, feasible and symmetric strut connectivities are obtained throug...
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A two-step topology-finding method based on mixed integer programming and nonlinear programming is proposed for tensegrity structures. In the first step, feasible and symmetric strut connectivities are obtained through a ground structure method combined with mixed integer programming;in the second step, the cable connectivities are optimized through nonlinear programming to obtain a feasible tensegrity structure. The same ground structure used in the first step is adopted in the second step, which is beneficial to find a more symmetric cable layout. The independent continuous mapping method is used in the second step to convert the 0-1 binary variables of cable connectivities to continuous variables to transform the combinatorial optimization problem into a nonlinear programming problem. The number of strut lengths is adopted as a control parameter and a symmetry objective function is proposed to generate a variety of regular and symmetric tensegrity structures. A multi-stage computation scheme is proposed to improve the computational efficiency. Typical examples are carried out to validate the proposed method. The computational efficiency of the method is benchmarked with existing methods fully based on mixed integer programming. Results demonstrate that the computational efficiency of the proposed method is significantly improved compared to the existing methods.
In this paper we propose a new local quasi-Newton method to solve the equality constrained nonlinear programming problem. The pivotal feature of the algorithm is that a projection of the Hessian of the Lagrangian is a...
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In this paper we propose a new local quasi-Newton method to solve the equality constrained nonlinear programming problem. The pivotal feature of the algorithm is that a projection of the Hessian of the Lagrangian is approximated by a sequence of symmetric positive definite matrices. The matrix approximation is updated at every iteration by a projected version of the DFP or BFGS formula: this involves two evaluations of the Lagrangian gradient per iteration. We establish that the method is locally convergent and the sequence of x-values converges to the solution at a 2-step Q-superlinear rate.
A simulation-based interval-fuzzy nonlinear programming (SIFNP) approach was developed for seasonal planning of stream water quality management. The techniques of inexact modeling, nonlinear programming, and interval-...
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A simulation-based interval-fuzzy nonlinear programming (SIFNP) approach was developed for seasonal planning of stream water quality management. The techniques of inexact modeling, nonlinear programming, and interval-fuzzy optimization were incorporated within a general framework. Based on a multi-segment stream water quality simulation model, dynamic waste assimilative capacity of a river system within a multi-season context was considered in the optimization process. The method could not only address complexities of various system uncertainties but also tackle nonlinear environmental-economic interrelationships in water quality management problems. In addition, interval-fuzzy numbers were introduced to reflect the dual uncertainties, i.e., imprecision associated with fixing the lower and upper bounds of membership functions. The proposed method was applied to a case of water quality management in the Guoyang section of the Guo River in China. Interval solutions reflecting the inherent uncertainties were generated, and a spectrum of cost-effective schemes for seasonal water quality management could thus be obtained by adjusting different combinations of the decision variables within their solution intervals. The results indicated that SIFNP could effectively communicate dual uncertainties into the optimization process and help decision makers to identify desired options under various complexities of system components.
For several types of finite or infinite dimensional optimization problems the marginal function (or optimal value function) is characterized by different local approximations such as generalized gradients, generalized...
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For several types of finite or infinite dimensional optimization problems the marginal function (or optimal value function) is characterized by different local approximations such as generalized gradients, generalized directional derivatives, directional Hadamard or Dini derivatives. We give estimates for these terms which are determined by multipliers satisfying necessary optimality conditions. When the functions which define the optimization problem are more than once continuously differentiable, then higher order necessary conditions are employed to obtain refined estimates for the marginal function. As a by-product we give a new equivalent formulation of Clarke's multiplier rule for nonsmooth optimization problems. This shows that the set of all multipliers satisfying these necessary conditions is the union of a finite number of closed convex cones.
作者：
Nie, Pu-YanJinan Univ
Dept Math Guangzhou 510632 Peoples R China Hunan Univ
Coll Econ & Trade Changsha 410079 Peoples R China
Penalty function methods, presented many years ago, play exceedingly important roles in the optimization community. According to numerical results, penalty function approaches work very efficiently for equality constr...
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Penalty function methods, presented many years ago, play exceedingly important roles in the optimization community. According to numerical results, penalty function approaches work very efficiently for equality constrained problems. For inequality constrained problems, sequential quadratic programming (SQP) approaches do better than that of sequential penalty quadratical programming (SlQP) methods. Taking these into account, we propose another optimization approach, in which we aim to combine the advantages of penalty function techniques and SQP approaches. In the new technique, equality constraints are handled by penalty function technique, while inequality constraints are still treated as constraints. The corresponding theories are exploited in this work. The theories of the corresponding augmented Lagrangian function, especially quadratic augmented penalty methods, are also achieved. A new kind of penalty method, combining the advantages of SQP and SlQP, is therefore developed in this work. (c) 2006 Elsevier Ltd. All rights reserved.
This paper proposes a new necessary condition for the infeasibility of nonlinear optimization problems, that becomes also sufficient under a convexity assumption, which is stated as a Pareto-criticality condition of a...
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This paper proposes a new necessary condition for the infeasibility of nonlinear optimization problems, that becomes also sufficient under a convexity assumption, which is stated as a Pareto-criticality condition of an auxiliary multi-objective optimization problem. This condition is evaluated in a search that either leads to a feasible point or to a point at which the infeasibility conditions hold. The resulting infeasibility certificate has global validity in convex problems and has at least a local meaning in generic nonlinear problems. (C) 2016 Elsevier B.V. All rights reserved.
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