共查询到20条相似文献,搜索用时 562 毫秒
1.
T. Antczak 《Journal of Optimization Theory and Applications》2018,176(1):205-224
In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given. 相似文献
2.
In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved. 相似文献
3.
Tadeusz Antczak 《Applied mathematics and computation》2011,217(15):6652-6662
A new exact penalty function method, called the l1 exact exponential penalty function method, is introduced. In this approach, the so-called the exponential penalized optimization problem with the l1 exact exponential penalty function is associated with the original optimization problem with both inequality and equality constraints. The l1 exact exponential penalty function method is used to solve nonconvex mathematical programming problems with r-invex functions (with respect to the same function η). The equivalence between sets of optimal solutions of the original mathematical programming problem and of its associated exponential penalized optimization problem is established under suitable r-invexity assumption. Also lower bounds on the penalty parameter are given, for which above these values, this result is true. 相似文献
4.
Tadeusz Antczak Marcin Studniarski 《Numerical Functional Analysis & Optimization》2016,37(12):1465-1487
In this article, the vector exact l1 penalty function method used for solving nonconvex nondifferentiable multiobjective programming problems is analyzed. In this method, the vector penalized optimization problem with the vector exact l1 penalty function is defined. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto optimal solutions of the considered nondifferentiable multiobjective programming problem and of the associated vector penalized optimization problem with the vector exact l1 penalty function. This equivalence is established for nondifferentiable invex vector optimization problems. Some examples of vector optimization problems are presented to illustrate the results established in the article. 相似文献
5.
Canghua Jiang Qun Lin Changjun Yu Kok Lay Teo Guang-Ren Duan 《Journal of Optimization Theory and Applications》2012,154(1):30-53
In this paper, we consider a class of optimal control problems with free terminal time and continuous inequality constraints. First, the problem is approximated by representing the control function as a piecewise-constant function. Then the continuous inequality constraints are transformed into terminal equality constraints for an auxiliary differential system. After these two steps, we transform the constrained optimization problem into a penalized problem with only box constraints on the decision variables using a novel exact penalty function. This penalized problem is then solved by a gradient-based optimization technique. Theoretical analysis proves that this penalty function has continuous derivatives, and for a sufficiently large and finite penalty parameter, its local minimizer is feasible in the sense that the continuous inequality constraints are satisfied. Furthermore, this local minimizer is also the local minimizer of the constrained problem. Numerical simulations on the range maximization for a hypersonic vehicle reentering the atmosphere subject to a heating constraint demonstrate the effectiveness of our method. 相似文献
6.
Tadeusz Antczak 《Optimization Letters》2016,10(7):1561-1576
In this paper, it is demonstrated that the exact absolute value penalty function method is useful for identifying the special sort of minimizers in nonconvex nonsmooth optimization problems with both inequality and equality constraints. The equivalence between the sets of strict global minima of order m in nonsmooth minimization problem and of its associated penalized optimization problem with the exact \(l_{1}\) penalty function is established under nondifferentiable \(\left( F,\rho \right) \)-convexity assumptions imposed on the involved functions. The threshold of the penalty parameter, above which this result holds, is also given. 相似文献
7.
Deriving the Properties of Linear Bilevel Programming via a Penalty Function Approach 总被引:7,自引:0,他引:7
Z. K. Xu 《Journal of Optimization Theory and Applications》1999,103(2):441-456
For the linear bilevel programming problem, we propose an assumption weaker than existing assumptions, while achieving similar results via a penalty function approach. The results include: equivalence between (i) existence of a solution to the problem, (ii) existence of an exact penalty function approach for solving the problem, and (iii) achievement of the optimal value of the equivalent form of the problem at some vertex of a certain polyhedral convex set. We prove that the assumption is both necessary and sufficient for the linear bilevel programming problem to admit an exact penalty function formulation, provided that the equivalent form of the problem has a feasible solution. A method is given for computing the minimal penalty function parameter value. This method can be executed by solving a set of linear programming problems. Lagrangian duality is also presented. 相似文献
8.
Bin Li Chang Jun Yu Kok Lay Teo Guang Ren Duan 《Journal of Optimization Theory and Applications》2011,151(2):260-291
In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous
state and control inequality constraints. By using the control parametrization technique and a time scaling transformation,
the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality
terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection
problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints.
On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems
with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed
from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives
rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value,
any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality
constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with
equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For
illustration, three examples are solved showing the effectiveness and applicability of the approach proposed. 相似文献
9.
An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems
Alexander J. Zaslavski 《Set-Valued and Variational Analysis》2011,19(3):385-398
We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces.
A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an
unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large
class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered
space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain
an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions
for the constrained minimization problems. 相似文献
10.
Alexander J. Zaslavski 《Set-Valued and Variational Analysis》2008,16(5-6):673-691
In this paper we use the penalty approach in order to study two constrained minimization problems. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property is stable under perturbations of cost functions, constraint functions and the right-hand side of constraints. 相似文献
11.
本文给出了广义可微精确罚函数的概念及一类所谓广义限域可微精确罚函数.本文预先选定罚因子,将不等式约束问题化为单一的无约束问题,并给出了具全局收敛性的算法.本文的罚函数构造简单,假设条件少而且算法的构造与收敛性结果是独特的. 相似文献
12.
在这篇文章中我们研究了对于不等式约束的非线性规划问题如何根据极小极大问题的鞍点来找精确罚问题的解。对于一个具有不等式约束的非线性规划问题,通过罚函数,我们构造出一个极小极大问题,应用交换“极小”或“极大”次序的策略,证明了罚问题的鞍点定理。研究结果显示极小极大问题的鞍点是精确罚问题的解。 相似文献
13.
M. V. Dolgopolik 《Journal of Optimization Theory and Applications》2018,176(3):728-744
In this two-part study, we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for constrained optimization problems in finite-dimensional spaces. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e., whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can be performed with the use of optimality conditions and constraint qualifications. In the first paper, we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle, we recover existing simple, necessary, and sufficient conditions for the global exactness of linear penalty functions and for the existence of augmented Lagrange multipliers of Rockafellar–Wets’ augmented Lagrangian. We also present completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems as well. 相似文献
14.
Zhiqing Meng Chuangyin Dang Xiaoqi Yang 《Computational Optimization and Applications》2006,35(3):375-398
In this paper we propose two methods for smoothing a nonsmooth square-root exact penalty function for inequality constrained
optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem,
of the nonsmooth penalty problem and of the original optimization problem. We develop an algorithm for solving the optimization
problem based on the smoothed penalty function and prove the convergence of the algorithm. The efficiency of the smoothed
penalty function is illustrated with some numerical examples, which show that the algorithm seems efficient. 相似文献
15.
Alexander J. Zaslavski 《Set-Valued Analysis》2007,15(3):223-237
In this paper we use the penalty approach in order to study constrained minimization problems in a complete metric space with
locally Lipschitzian mixed constraints. A penalty function is said to have the exact penalty property if there is a penalty
coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem.
In this paper we establish sufficient conditions for the exact penalty property.
相似文献
16.
17.
In high-dimensional and/or nonparametric regression problems , regularization (or penalization) is used to control model complexity and induce desired structure. Each penalty has a weight parameter that indicates how strongly the structure corresponding to that penalty should be enforced. Typically, the parameters are chosen to minimize the error on a separate validation set using a simple grid search or a gradient-free optimization method. It is more efficient to tune parameters if the gradient can be determined, but this is often difficult for problems with nonsmooth penalty functions. Here, we show that for many penalized regression problems, the validation loss is actually smooth almost-everywhere with respect to the penalty parameters. We can, therefore, apply a modified gradient descent algorithm to tune parameters. Through simulation studies on example regression problems, we find that increasing the number of penalty parameters and tuning them using our method can decrease the generalization error. 相似文献
18.
This paper proposes a mechanism to produce equivalent Lipschitz surrogates for zero-norm and rank optimization problems by means of the global exact penalty for their equivalent mathematical programs with an equilibrium constraint (MPECs). Specifically, we reformulate these combinatorial problems as equivalent MPECs by the variational characterization of the zero-norm and rank function, show that their penalized problems, yielded by moving the equilibrium constraint into the objective, are the global exact penalization, and obtain the equivalent Lipschitz surrogates by eliminating the dual variable in the global exact penalty. These surrogates, including the popular SCAD function in statistics, are also difference of two convex functions (D.C.) if the function and constraint set involved in zero-norm and rank optimization problems are convex. We illustrate an application by designing a multi-stage convex relaxation approach to the rank plus zero-norm regularized minimization problem. 相似文献
19.
Xinsheng Xu Chuangyin Dang Felix T. S. Chan Yongli Wang 《Numerical Functional Analysis & Optimization》2019,40(1):1-18
This article introduces a smoothing technique to the l1 exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm. 相似文献
20.
In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American
options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem
is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown
that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order.
This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter.
A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the
theoretical findings and to show the effectiveness and usefulness of the method.
This work was partially supported by a research grant from the University of Western Australia and the Research Grant Council
of Hong Kong, Grants PolyU BQ475 and PolyU BQ493. 相似文献