Combined phase I—phase II methods of feasible directions

“…The maps A 1 and A 2 are defined as follows. For all z in R", all e>0, the first order search direction %(z) is defined [9] to be the solution of the quadratic program:…”

Section: A Stabilised Conceptual Algorithmmentioning

confidence: 99%

A quadratically convergent algorithm for solving infinite dimensional inequalities

Mayne

Polak

1982

Appl Math Optim

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“…If x is a feasible point, it is clear that (x) = 0, and then (1.3) reduces to the direction finding problem of Topkis and Veinott's feasible direction method [26]. Because their inner feasible direction algorithms only use the information of first derivatives, these phase I-phase II algorithms in [23] converge linearly at best. In order to increase the speed of convergence, combining the ideas from [23] and feasible SQP, Jian [12,Ch.…”

Section: Introductionmentioning

confidence: 95%

“…So roughly speaking, the goal of this paper is to present a unified algorithm that well combines the phase of computing a feasible point and the phase of a feasible SQP algorithm. An early version of phase I-phase II method we mention here for solving problem (P) was proposed by Polak et al [23]. Their algorithms can automatically produce a feasible starting point and then perform the operation of feasible direction algorithm to get an optimal solution.…”

Section: Introductionmentioning

confidence: 99%

A New Superlinearly Convergent Strongly Subfeasible Sequential Quadratic Programming Algorithm for Inequality-Constrained Optimization

Jian¹,

Tang²,

Hu³

et al. 2008

Numerical Functional Analysis and Optimization

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Combining the ideas of generalized projection and the strongly subfeasible sequential quadratic programming (SQP) method, we present a new strongly subfeasible SQP algorithm for nonlinearly inequality-constrained optimization problems. The algorithm, in which a new unified step-length search of Armijo type is introduced, starting from an arbitrary initial point, produces a feasible point after a finite number of iterations and from then on becomes a feasible descent SQP algorithm. At each iteration, only one quadratic program needs to be solved, and two correctional directions are obtained simply by explicit formulas that contain the same inverse matrix. Furthermore, the global and superlinear convergence results are proved under mild assumptions without strict complementarity conditions. Finally, some preliminary numerical results show that the proposed algorithm is stable and promising.

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“…However, the MFD always requires a feasible initial point, so an auxiliary procedure must be considered for obtaining such a starting point, further sometimes solving the auxiliary problem may be difficult as solving the original one itself. To overcome this shortcoming, Polak, Trahan, and Mayne [11] presented a combined Phase I-Phase II algorithm with an arbitrary initial point in 1979. The algorithm becomes an MFD when the iterative point gets into the feasible set.…”

Section: Introductionmentioning

confidence: 98%

A new ɛ-generalized projection method of strongly sub-feasible directions for inequality constrained optimization

Jian

Guo

2011

J Syst Sci Complex

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In this paper, the nonlinear optimization problems with inequality constraints are discussed. Combining the ideas of the strongly sub-feasible directions method and the ε-generalized projection technique, a new algorithm starting with an arbitrary initial iteration point for the discussed problems is presented. At each iteration, the search direction is generated by a new ε-generalized projection explicit formula, and the step length is yielded by a new Armijo line search. Under some necessary assumptions, not only the algorithm possesses global and strong convergence, but also the iterative points always get into the feasible set after finite iterations. Finally, some preliminary numerical results are reported.

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Combined phase I—phase II methods of feasible directions

Cited by 75 publications

References 8 publications

A quadratically convergent algorithm for solving infinite dimensional inequalities

A quadratically convergent algorithm for solving infinite dimensional inequalities

A New Superlinearly Convergent Strongly Subfeasible Sequential Quadratic Programming Algorithm for Inequality-Constrained Optimization

A new ɛ-generalized projection method of strongly sub-feasible directions for inequality constrained optimization

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