A restricted trust region algorithm for unconstrained optimization

Bulteau, Julie; Vial, Jean–Philippe

doi:10.1007/bf00942189

Cited by 16 publications

(7 citation statements)

References 19 publications

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“…To see this, we need the following lemma, which states the optimality system of (SP). This is a simple extension of the known result for problems without equality constraints; see [6].…”

Section: The Algorithmmentioning

confidence: 76%

A Trust Region Interior Point Algorithm for Linearly Constrained Optimization

Bonnans¹,

Pola²

1997

SIAM J. Optim.

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Abstract. We present an extension, for nonlinear optimization under linear constraints, of an algorithm for quadratic programming using a trust region idea introduced by Ye and Tse [Math. Programming, 44 (1989), pp. 157-179] and extended by Bonnans and Bouhtou [RAIRO Rech. Opér., 29 (1995), pp. 195-217]. Due to the nonlinearity of the cost, we use a linesearch in order to reduce the step if necessary. We prove that, under suitable hypotheses, the algorithm converges to a point satisfying the first-order optimality system, and we analyze under which conditions the unit stepsize will be asymptotically accepted.Key words. trust region, quadratic model, linesearch, interior points AMS subject classifications. 90C30, 65K05, 49M40 PII. S10526234932506391. Introduction. In this paper, we study an algorithm for minimizing a nonlinear cost under linear constraints. We consider problems with linear equality constraints and nonnegative variables. At each step, a direction is computed by minimizing a convex quadratic model over an ellipsoidal trust region, and then a linesearch of Armijo type is performed in this direction. At each iteration, the ellipsoid of the quadratic problem is so small that it forces the nonnegativity constraints to be satisfied. However, the ellipsoid is not necessarily contained in the set of feasible points.In the case of linear programming (LP) or convex quadratic programming (QP), we may assume the quadratic model to be equal to the cost function. Then the unit step will be accepted by the linesearch. In the case of LP, the algorithm is then reduced to the celebrated Dikin's algorithm [10] (see also Tsuchiya [26]). Ye and Tse [27] have extended this algorithm to convex quadratic programming using the trust region idea. This problem was also considered by Sun [25]. Bonnans and Bouhtou [2] studied such methods for nonconvex quadratic problems by taking a variable size for the trust region. An early extension of trust region algorithms to nonlinear costs is done in Dikin and Zorkalcev [11]. Among the related work, we quote Gonzaga and Carlos [13]. Interior point algorithms for the solution of constrained convex optimization problems have been studied by many other researchers; see, for instance, Den Hertog, Roos, and Terlaky [8] Dennis, Heinkenschloss, and Vicente [9], and Coleman and Li [7]. Gonzaga [14] explores the shape of the trust regions to generate ellipsoidal regions adapted to the shape of the feasible set. The resulting algorithm generates sequences of points in the interior of the feasible set.In this paper, we obtain some results of global convergence, comparable to those obtained in [2] for QP; by global convergence we only mean that the limit points of the sequence generated by the algorithm satisfy the first-order optimality system. The main novelty of the paper, however, is in the local analysis in the vicinity of

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“…To see this, we need the following lemma, which states the optimality system of (SP). This is a simple extension of the known result for problems without equality constraints; see [6].…”

Section: The Algorithmmentioning

confidence: 76%

A Trust Region Interior Point Algorithm for Linearly Constrained Optimization

Bonnans¹,

Pola²

1997

SIAM J. Optim.

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“…In particular, it should contain the gradient line and the Newton direction or a direction of negative curvature, if it exists. This idea is expanded in the papers of Bulteau and Vial [3,4]. For a more readable proof, we state the following intermediary result as a lemma.…”

Section: Resultsmentioning

confidence: 99%

“…In particular, we show that, with the approximate gradient path, the convergence results are about as strong as those obtained with the optimal path. Finally, the present analysis may serve as a framework for devising new paths that would be both easily implementable and endowed with interesting properties (BuReau and Vial [3,4]). …”

Section: Introductionmentioning

confidence: 99%

Curvilinear path and trust region in unconstrained optimization: A convergence analysis

Bulteau

Vial

1987

Mathematical Programming Studies

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In this paper we propose a general algorithm for solving unconstrained optimization problems. The basic step of the algorithm consists in finding a "good" successor point to the current iterate by choosing it along a curvilinear path and within a trust region. This scheme is due to Powell and it has been applied by Sorensen to a particular type of path. We give a series of properties that an arbitrary path should satisfy in order to achieve global convergence and fast asymptotic convergence. We review various paths that have been proposed in the literature and study the extent to which they satisfy our properties.

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“…As X^2 is positive definite, there exists a threshold value F& > 0 (we do not consider négative values of v) such that i/j k (x) is convex on Af(A), if and only if v > v k , Also by (2.5) this v k satisfies v^ <v k , By classical argument [7], [18] one can prove that the function v -> (ar* -…”

Section: The Extended Algorithmmentioning

confidence: 99%

“…The proof uses the optimality system of (SP) stated below. It is a simple extension of the known resuit for problems without equality constraints, see [7], [18]. …”

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confidence: 99%