On Some Properties of Quadratic Programs with a Convex Quadratic Constraint

Lucidi, Stefano; Palagi, Laura; Roma, Massimo

doi:10.1137/s1052623494278049

Cited by 44 publications

(35 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…M. Martinez [15] has investigated the nature of local nonglobal solutions of (Q 1 ) and (Q 2 ) and has shown the following interesting property (especially for local algorithms): these problems have at most one local nonglobal solution. Moreover, being inspired by G. E. Forsythe and G. H. Golub's work [3], S. Lucidi, L. Palagi, and M. Roma [13] have stated a very nice result: in (Q 1 ) the objective function can admit at most 2m + 2 different values at Kuhn-Tucker points, where m is the number of distinct negative eigenvalues of A.…”

Section: Introductionmentioning

confidence: 99%

A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem

Tao¹,

Thi²

1998

SIAM J. Optim.

468

135

View full text Add to dashboard Cite

Abstract. This paper is devoted to difference of convex functions (d.c.) optimization: d.c. duality, local and global optimality conditions in d.c. programming, the d.c. algorithm (DCA), and its application to solving the trust-region problem. The DCA is an iterative method that is quite different from well-known related algorithms. Thanks to the particular structure of the trust-region problem, the DCA is very simple (requiring only matrix-vector products) and, in practice, converges to the global solution. The inexpensive implicitly restarted Lanczos method of Sorensen is used to check the optimality of solutions provided by the DCA. When a nonglobal solution is found, a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the DCA at this point. It is shown that in the nonconvex case, the DCA converges to the global solution of the trust-region problem, using only matrix-vector products and requiring at most 2m + 2 restarts, where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem. Numerical simulations establish the robustness and efficiency of the DCA compared to standard related methods, especially for large-scale problems.

show abstract

Section: Introductionmentioning

confidence: 99%

A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem

Tao¹,

Thi²

1998

SIAM J. Optim.

468

135

View full text Add to dashboard Cite

show abstract

“…A particular case of (15) is the classical trust-region subproblem, where f is quadratic. Recently (see [20,25]) procedures for escaping from nonglobal stationary points of this problem have been found, and so it becomes increasingly important to obtain fast algorithms for finding critical points, especially in the large-scale case. (See [28,29,31].…”

Section: Final Remarksmentioning

confidence: 99%

Nonmonotone Spectral Projected Gradient Methods on Convex Sets

Birgin

Martínez

Raydan³

2000

SIAM J. Optim.

927

698

View full text Add to dashboard Cite

Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo-Lampariello-Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the one-dimensional search process. Convergence properties and extensive numerical results are presented.Key words. projected gradients, nonmonotone line search, large-scale problems, bound constrained problems, spectral gradient method AMS subject classifications. 49M07, 49M10, 65K, 90C06, 90C20 PII. S10526234973309631. Introduction. We consider the projected gradient method for the minimization of differentiable functions on nonempty closed and convex sets. Over the last few decades, there have been many different variations of the projected gradient method that can be viewed as the constrained extensions of the optimal gradient method for unconstrained minimization. They all have the common property of maintaining feasibility of the iterates by frequently projecting trial steps on the feasible convex set. This process is in general the most expensive part of any projected gradient method. Moreover, even if projecting is inexpensive, as in the box-constrained case, the method is considered to be very slow, as is its analogue, the optimal gradient method (also known as steepest descent), for unconstrained optimization. On the positive side, the projected gradient method is quite simple to implement and very effective for large-scale problems.This state of affairs motivates us to combine the projected gradient method with two recently developed ingredients in optimization. First we extend the typical globalization strategies associated with these methods to the nonmonotone line search schemes developed by Grippo, Lampariello, and Lucidi [17] for Newton's method. Second, we propose to associate the spectral steplength, introduced by Barzilai and Borwein [1] and analyzed by Raydan [26]. This choice of steplength requires little computational work and greatly speeds up the convergence of gradient methods. In fact, while the spectral gradient method appears to be a generalized steepest descent method, it is clear from its derivation that it is related to the quasi-Newton family of methods through an approximated secant equation. The fundamental difference is

show abstract

“…In the survey [59], Palagi addresses other possibilities for the numerical solution of the largescale TRS: the parametric eigenvalue reformulation-based strategy of Sorensen [80]; the semidefinite programming approach of Rendl & Wolkowicz [70]; the exact penalty function based algorithm of Lucidi, Palagi & Roma [46], and the DC (difference of convex functions) based algorithm of Pham Dinh Tao & Le Thi Hoai An [82].…”

Section: Trust-region Subproblemsmentioning

confidence: 99%

Trust-Region-Based Methods for Nonlinear Programming: Recent Advances and Perspectives

Santos

2014

Pesqui. Oper.

View full text Add to dashboard Cite

ABSTRACT. The aim of this text is to highlight recent advances of trust-region-based methods for nonlinear programming and to put them into perspective. An algorithmic framework provides a ground with the main ideas of these methods and the related notation. Specific approaches concerned with handling the trustregion subproblem are recalled, particularly for the large scale setting. Recent contributions encompassing the trust-region globalization technique for nonlinear programming are reviewed, including nonmonotone acceptance criteria for unconstrained minimization; the adaptive adjustment of the trust-region radius; the merging of the trust-region step into a line search scheme, and the usage of the trust-region elements within derivative-free optimization algorithms.

show abstract

On Some Properties of Quadratic Programs with a Convex Quadratic Constraint

Cited by 44 publications

References 37 publications

A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem

A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem

Nonmonotone Spectral Projected Gradient Methods on Convex Sets

Trust-Region-Based Methods for Nonlinear Programming: Recent Advances and Perspectives

Contact Info

Product

Resources

About