Search

For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well defined and necessarily converges to this specific solution (despite degeneracy, and despite that there are other solutions nearby). We note that unlike the common settings of convergence analyses, our assumptions subsume that a local Lipschitzian error bound does not hold for the solution in question. Our results apply to constrained and projected variants of the Gauss-Newton, Levenberg-Marquardt, and LP-Newton methods. Applications to smooth and piecewise smooth reformulations of complementarity problems are also discussed. [ABSTRACT FROM AUTHOR]

This paper proposes a self-adaptive penalty function and presents a penalty-based algorithm for solving nonsmooth and nonconvex constrained optimization problems. We prove that the general constrained optimization problem is equivalent to a bound constrained problem in the sense that they have the same global solutions. The global minimizer of the penalty function subject to a set of bound constraints may be obtained by a population-based meta-heuristic. Further, a hybrid self-adaptive penalty firefly algorithm, with a local intensification search, is designed, and its convergence analysis is established. The numerical experiments and a comparison with other penalty-based approaches show the effectiveness of the new self-adaptive penalty algorithm in solving constrained global optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we establish the convergence properties for a majorized alternating direction method of multipliers for linearly constrained convex optimization problems, whose objectives contain coupled functions. Our convergence analysis relies on the generalized Mean-Value Theorem, which plays an important role to properly control the cross terms due to the presence of coupled objective functions. Our results, in particular, show that directly applying two-block alternating direction method of multipliers with a large step length of the golden ratio to the linearly constrained convex optimization problem with a quadratically coupled objective function is convergent under mild conditions. We also provide several iteration complexity results for the algorithm. [ABSTRACT FROM AUTHOR]

An iteration of the stabilized sequential quadratic programming method consists in solving a certain quadratic program in the primal-dual space, regularized in the dual variables. The advantage with respect to the classical sequential quadratic programming is that no constraint qualifications are required for fast local convergence (i.e., the problem can be degenerate). In particular, for equality-constrained problems, the superlinear rate of convergence is guaranteed under the only assumption that the primal-dual starting point is close enough to a stationary point and a noncritical Lagrange multiplier (the latter being weaker than the second-order sufficient optimality condition). However, unlike for the usual sequential quadratic programming method, designing natural globally convergent algorithms based on the stabilized version proved quite a challenge and, currently, there are very few proposals in this direction. For equality-constrained problems, we suggest to use for the task linesearch for the smooth two-parameter exact penalty function, which is the sum of the Lagrangian with squared penalizations of the violation of the constraints and of the violation of the Lagrangian stationarity with respect to primal variables. Reasonable global convergence properties are established. Moreover, we show that the globalized algorithm preserves the superlinear rate of the stabilized sequential quadratic programming method under the weak conditions mentioned above. We also present some numerical experiments on a set of degenerate test problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose an active set modified Polak-Ribiére-Polyak method for solving large-scale optimization with simple bounds on the variables. The active set is guessed by an identification technique at each iteration and the recently developed modified Polak-Ribiére-Polyak method is used to update the variables with indices outside of the active set. Under appropriate conditions, we show that the proposed method is globally convergent. Numerical experiments are presented using box constrained problems in the CUTEr test problem libraries. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed. [ABSTRACT FROM AUTHOR]

In this paper, we adapt a genetic algorithm for constrained optimization problems. We use a dynamic penalty approach along with some form of annealing, thus forcing the search to concentrate on feasible solutions as the algorithm progresses. We suggest two different general-purpose methods for guaranteeing convergence to a globally optimal (feasible) solution, neither of which makes any assumptions on the structure of the optimization problem. The former involves modifying the GA evolution operators to yield a Boltzmann-type distribution on populations. The latter incorporates a dynamic penalty along with a slow annealing of acceptance probabilities. We prove that, with probability one, both of these methods will converge to a globally optimal feasible state. [ABSTRACT FROM AUTHOR]

We propose a pattern search method to solve a classical nonsmooth optimization problem. In a deep analogy with pattern search methods for linear constrained optimization, the set of search directions at each iteration is defined in such a way that it conforms to the local geometry of the set of points of nondifferentiability near the current iterate. This is crucial to ensure convergence. The approach presented here can be extended to wider classes of nonsmooth optimization problems. Numerical experiments seem to be encouraging. [ABSTRACT FROM AUTHOR]

In this note, we prove that the convergence results for vector optimization problems with equilibrium constraints presented in Wu and Cheng (J. Optim. Theory Appl. 125, 453–472, 2005) are not correct. Actually, we show that results of this type cannot be established at all. This is due to the possible lack, even under nice assumptions, of lower convergence of the solution map for equilibrium problems, already deeply investigated in Loridan and Morgan (Optimization 20, 819–836, 1989) and Lignola and Morgan (J. Optim. Theory Appl. 93, 575–596, 1997). [ABSTRACT FROM AUTHOR]

This paper proposes a line search technique to satisfy a relaxed form of the strong Wolfe conditions in order to guarantee the descent condition at each iteration of the Polak-Ribière-Polyak conjugate gradient algorithm. It is proved that this line search algorithm preserves the usual convergence properties of any descent algorithm. In particular, it is shown that the Zoutendijk condition holds under mild assumptions. It is also proved that the resulting conjugate gradient algorithm is convergent under a strong convexity assumption. For the nonconvex case, a globally convergent modification is proposed. Numerical tests are presented. [ABSTRACT FROM AUTHOR]