A Strongly Convergent Method for Nonsmooth Convex Minimization in Hilbert Spaces

Oliveira

2015

Optimization

A key procedure in proximal bundle methods for convex minimization problems is the definition of stability centres, which are points generated by the iterative process that successfully decrease the objective function. In this paper we study a different stability-centre classification rule for proximal bundle methods. We show that the proposed bundle variant has at least two particularly interesting features: (i) the sequence of stability centres generated by the method converges strongly to the solution that lies closest to the initial point; (ii) if the sequence of stability centres is finite,x being its last element, then the sequence of nonstability centres (null steps) converges strongly tox. Property (i) is useful in some practical applications in which a minimal norm solution is required. We show the interest of this property on several instances of a full sized unit-commitment problem.

Section: Introductionmentioning

confidence: 90%

Section: Notationmentioning

confidence: 99%

A strongly convergent proximal bundle method for convex minimization in Hilbert spaces

Ackooij¹,

Oliveira

2015

Optimization

“…We emphasize that Assumption (A1) is a typical hypothesis for proving the convergence of the subgradient-scalar methods in infinite dimension setting; see [1,8,9,25]. As stated in [23], for the scalar and vector framework, this assumption holds trivially in finite-dimensional spaces.…”

Section: Assumptionsmentioning

confidence: 99%

“…The proposed conceptual algorithm has two variants called Algorithm R and Algorithm S. The first one is based on Robinson's subgradient algorithm given in [27] for solving problem (4). The S variant corresponds to a special modification of the subgradient algorithms proposed in [9] for the scalar problem (m = 1 and K = Ê + ) and in [10] for solving problem (4). The main difference between the proposed variants lies in how the projection step is done.…”

Section: Introductionmentioning

confidence: 99%

Subgradient algorithms for solving variable inequalities

Applied Mathematics and Computation

Allende

Pérez

2014

a b s t r a c tIn this paper we consider the variable inequalities problem, that is, to find a solution of the inclusion given by the sum of a function and a point-to-cone application. This problem can be seen as a generalization of the classical inequalities problem taking a variable order structure. Exploiting this relation, we propose two variants of the subgradient algorithm for solving the variable inequalities model. The convergence analysis is given under convex-like conditions, which, when the point-to-cone application is constant, contains the old subgradient schemes.

“…This kind of hyperplane has been used in some works, see [24,25]. The variants of the conceptual algorithm given in (12), (13) and (14) are called Algorithm 1, 2 and 3, respectively.…”

Section: Iterative Step 2: Setmentioning

confidence: 99%

A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy

Millán²

2014

Optimization

In this paper, we propose variants of Forward-Backward splitting method for finding a zero of the sum of two operators. A classical modification of ForwardBackward method was proposed by Tseng, which is known to converge when the forward and the backward operators are monotone and with Lipschitz continuity of the forward operator. The conceptual algorithm proposed here improves Tseng's method in some instances. The first and main part of our approach, contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. During the iteration process, the search performs only one calculation of the forward-backward operator in each tentative of the step. This achieves a considerable computational saving when the forwardbackward operator is computationally expensive. The second part of the scheme consists in special projection steps. The convergence analysis of the proposed scheme is given assuming monotonicity on both operators, without Lipschitz continuity assumption on the forward operator.