A Truncated Projected Newton-Type Algorithm for Large-Scale Semi-infinite Programming

Qin, Ni; Ling, Chen; Qi, Liqun; Teo, Kok Lay

doi:10.1137/040619867

Cited by 22 publications

(11 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, the constraint violation improves as we increase the number of samples. When N is increased from 100 to 1000, the improvement is not significant, suggesting that N = 100 is a sufficiently large sample size for Problem (40).…”

Section: Numerical Experimentsmentioning

confidence: 98%

An inexact primal-dual algorithm for semi-infinite programming

2020

View full text Add to dashboard Cite

This paper considers an inexact primal-dual algorithm for semi-infinite programming (SIP) for which it provides general error bounds. To implement the dual variable update, we create a new prox function for nonnegative measures which turns out to be a generalization of the Kullback-Leibler divergence for probability distributions. We show that under suitable conditions on the error, this algorithm achieves an O(1/ √ K) rate of convergence in terms of the optimality gap and constraint violation. We then use our general error bounds to analyze the convergence and sample complexity of a specific primal-dual SIP algorithm based on Monte Carlo integration. Finally, we provide numerical experiments to demonstrate the performance of our algorithm. a certain computationally-cheap criterion. In [39], the earlier central cutting plane algorithm from [29] is extended to allow for nonlinear convex cuts.Randomized cutting plane algorithms have recently been developed for SIP in [6,7,14]. The idea is to input a probability distribution over the constraints, randomly sample a modest number of constraints, and then solve the resulting relaxed problem. Intuitively, as long as a sufficient number of samples of the constraints is drawn, the resulting randomized solution should violate only a small portion of the constraints and achieve near-optimality.Discretization methods: In the discretization approach, a sequence of relaxed problems with a finite number of constraints is solved according to a predefined or adaptively controlled grid generation scheme [46,49]. Discretization methods are generally computationally expensive. The convergence rate of the error between the solution of the SIP problem and the solution of the discretized program is investigated in [49].Local reduction methods: In the local reduction approach, an SIP problem is reduced to a problem with a finite number of constraints [21]. The reduced problem involves constraints which are defined only implicitly, and the resulting problem is solved via the Newton method which has good local convergence properties. However, local reduction methods require strong assumptions and are often conceptual.Dual methods: A wide class of SIP algorithms is based on directly solving the KKT conditions. In [28,33,34], the authors derive Wolfe's dual for an SIP and discuss numerical schemes for this problem. The KKT conditions often have some degree of smoothness, and so various Newton-type methods can be applied [45,30,40,44]. However, feasibility is not guaranteed under the all Newton-type methods. A new smoothing Newton-type method is proposed to overcome this drawback in [32].Applications: SIP is the basis of the approximate linear programming (ALP) approach for dynamic programming. In [16,3,13], the authors consider various schemes for solving ALPs. The idea of random constraint sampling in ALP is developed in [16,3]. In [31], an adaptive constraint sampling approach called 'ALP-Secant' is proposed which is based on solving a sequence of saddle-point problems. It is shown that A...

show abstract

Section: Numerical Experimentsmentioning

confidence: 98%

An inexact primal-dual algorithm for semi-infinite programming

2020

View full text Add to dashboard Cite

show abstract

“…Based on the KKT system of SIP, some Newton-type methods are presented (see [11,13,16,23]). In our recent work [22], we presented a smoothing quasi-Newton algorithm for the reformulated SIP problem.…”

Section: Iterative Approachmentioning

confidence: 99%

“…Hence, our attention in this research is to design efficient numerical methods for OTS, combing with the special structure of the reformulated SIP and the new approach in SIP problems. Motivated by the recently works for SIP problems (see [6,7,17,20,21] and reference therein), especially for the recent Newton-type methods [11,13,16,23], we develop the method proposed in [23] for solving the OTS problems. We first reformulate the OTS problem to a SIP problem by using a similar way in [1].…”

Section: Introductionmentioning

confidence: 99%

New approach for the nonlinear programming with transient stability constraints arising from power systems

Tong

Zhou

2008

Comput Optim Appl

View full text Add to dashboard Cite

show abstract

“…In recent years, (generalized) semi-infinite optimization has become an active field of research in applied mathematics; see the survey paper [3] as well as a book [2] which contains, in tutorial form, several survey papers on (generalized) semi-infinite programming and other related topics such as semi-definite programming, optimal control, etc. For example, for standard semi-infinite programming, some first-order optimality condition in terms of firstorder multipliers have been established [4,10,11,14,15], whereas many efficient algorithms such as central cutting plane method, Newton method, SQP method, maximum entropy method and nonlinear penalty method have been proposed [1,5,9,12,13]. The fact that the infinite index set Y (x) depends on variable x in (GSMMP) makes the problem complicated significantly.…”

Section: Introductionmentioning

confidence: 99%

Equivalent Reformulations of Generalized Semi-infinite Min-Max Programming

Zhou

Zhao²,

Xu³

2008

2008 ISECS International Colloquium on Computing, Communication, Control, and Management

View full text Add to dashboard Cite

New equivalent reformulations of generalized semiinfinite min-max programming (GSMMP) is derived by addressing the perturbed problem of the lower-level programming. Based on these, we obtain an exact penalty representations for the lower-level programming, which allows us to convert (GSSMP) into the standard semi-infinite programming problems (SMMP) via a class of exact penalty functions, including not only the function used in [9] but also some smooth exact penalty functions as special cases. This fact makes it possible to solve (GSMMP) by using any available standard semi-infinite optimization algorithms.

show abstract

A Truncated Projected Newton-Type Algorithm for Large-Scale Semi-infinite Programming

Cited by 22 publications

References 16 publications

An inexact primal-dual algorithm for semi-infinite programming

An inexact primal-dual algorithm for semi-infinite programming

New approach for the nonlinear programming with transient stability constraints arising from power systems

Equivalent Reformulations of Generalized Semi-infinite Min-Max Programming

Contact Info

Product

Resources

About