An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

Journal of Optimization Theory and Applications - A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type...     

An Augmented Lagrangian Method Exploiting an Active-Set Strategy and Second-Order Information

Journal of Optimization Theory and Applications - In this paper, we consider nonlinear optimization problems with nonlinear equality constraints and bound constraints on the variables. For the...     

Partial Augmented Lagrangian Method and Mathematical Programs with Complementarity Constraints

In this paper, we apply a partial augmented Lagrangian method to mathematical programs with complementarity constraints (MPCC). Specifically, only the complementarity constraints are incorporated into the objective function of the augmented Lagrangian problem while the other constraints of the original MPCC are retained as constraints in the augmented Lagrangian problem. We show that the limit point of a sequence of points that satisfy second-order necessary conditions of the partial augmented Lagrangian problems is a strongly stationary point (hence a B-stationary point) of the original MPCC if the limit point is feasible to MPCC, the linear independence constraint qualification for MPCC and the upper level strict complementarity condition hold at the limit point. Furthermore, this limit point also satisfies a second-order necessary optimality condition of MPCC. Numerical experiments are done to test the computational performances of several methods for MPCC proposed in the literature. This research was partially supported by the Research Grants Council (BQ654) of Hong Kong and the Postdoctoral Fellowship of The Hong Kong Polytechnic University. Dedicated to Alex Rubinov on the occassion of his 65th birthday.     

条形装箱问题的优化模型和增广Lagrange算法

This paper formulates a two-dimensional strip packing problem as a non-linear programming(NLP)problem and establishes the first-order optimality con-ditions for the NLP problem.A numerical algorithm for solving this NLP problemis given to find exact solutions to strip-packing problems involving up to 10 items.Approximate solutions can be found for big-sized problems by decomposing the setof items into small-sized blocks of which each block adopts the proposed numericalalgorithm.Numerical results show that the approximate solutions to big-sized prob-lems obtained by this method are superior to those by NFDH,FFDH and BFDHapproaches.     

Viability Theorem for Deterministic Mean Field Type Control Systems

A mean field type control system is a dynamical system in the Wasserstein space describing an evolution of a large population of agents with mean-field interaction under a control of a unique decision maker. We develop the viability theorem for the mean field type control system. To this end we introduce a set of tangent elements to the given set of probabilities. Each tangent element is a distribution on the tangent bundle of the phase space. The viability theorem for mean field type control systems is formulated in the classical way: the given set of probabilities on phase space is viable if and only if the set of tangent distributions intersects with the set of distributions feasible by virtue of dynamics.     

Parameter Estimation with the Augmented Lagrangian Method for a Parabolic Equation

In this paper, we investigate the numerical identification of the diffusion parameters in a linear parabolic problem. The identification is formulated as a constrained minimization problem. By using the augmented Lagrangian method, the inverse problem is reduced to a coupled nonlinear algebraic system, which can be solved efficiently with the preconditioned conjugate gradient method. Finally, we present some numerical experiments to show the efficiency of the proposed methods, even for identifying highly discontinuous parameters.This work was partially supported by the Research Council of Norway, Grant NFR-128224/431.     

An Exact Augmented Lagrangian Function for Nonlinear Programming with Two-Sided Constraints

This paper is aimed toward the definition of a new exact augmented Lagrangian function for two-sided inequality constrained problems. The distinguishing feature of this augmented Lagrangian function is that it employs only one multiplier for each two-sided constraint. We prove that stationary points, local minimizers and global minimizers of the exact augmented Lagrangian function correspond exactly to KKT pairs, local solutions and global solutions of the constrained problem.     

An Augmented Lagrangian based Semismooth Newton Method for a Class of Bilinear Programming Problems

This paper proposes a semismooth Newton method for a class of bilinear programming problems(BLPs) based on the augmented Lagrangian, in which the BLPs are reformulated as a system of nonlinear equations with original variables and Lagrange multipliers. Without strict complementarity, the convergence of the method is studied by means of theories of semismooth analysis under the linear independence constraint qualification and strong second order sufficient condition. At last, numerical results are reported to show the performance of the proposed method.     

Perturbed Augmented Lagrangian Method Framework with Applications to Proximal and Smoothed Variants

Journal of Optimization Theory and Applications - We introduce a perturbed augmented Lagrangian method framework, which is a convenient tool for local analyses of convergence and rates of...     

Updating the Multipliers Associated with Inequality Constraints in an Augmented Lagrangian Multiplier Method

This paper contributes to the development of the field of augmented Lagrangian multiplier methods for general nonlinear programming by introducing a new update for the multipliers corresponding to inequality constraints. The update maintains naturally the nonnegativity of the multipliers without the need for a positive-orthant projection, as a result of the verification of the first-order necessary conditions for the minimization of a modified augmented Lagrangian penalty function.In the new multiplier method, the roles of the multipliers are interchanged: the multipliers corresponding to the inequality constraints are updated explicitly, whereas the multipliers corresponding to the equality constraints are approximated implicitly. It is shown that the basic properties of local convergence of the traditional multiplier method are valid also for the proposed method.     

On an Augmented Lagrangian SQP Method for a Class of Optimal Control Problems in Banach Spaces

An augmented Lagrangian SQP method is discussed for a class of nonlinear optimal control problems in Banach spaces with constraints on the control. The convergence of the method is investigated by its equivalence with the generalized Newton method for the optimality system of the augmented optimal control problem. The method is shown to be quadratically convergent, if the optimality system of the standard non-augmented SQP method is strongly regular in the sense of Robinson. This result is applied to a test problem for the heat equation with Stefan-Boltzmann boundary condition. The numerical tests confirm the theoretical results.     

Control and Nash Games with Mean Field Effect

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007, of Gueant-Lasry-Lions in 2011, of Huang-Caines-Malham in 2007 and many others). There are a lot of applications. In general, the applications concern approximating an infinite number of players with common behavior by a representative agent. This agent has to solve a control problem perturbed by a field equation, representing in some way the behavior of the average infinite number of agents. This approach does not lead easily to the problems of Nash equilibrium for a finite number of players, perturbed by field equations, unless one considers averaging within different groups, which has not been done in the literature, and seems quite challenging. In this paper, the authors approach similar problems with a different motivation which makes sense for control and also for differential games. Thus the systems of nonlinear partial differential equations with mean field terms, which have not been addressed in the literature so far, are considered here.     

Fast Augmented Lagrangian Method in the convex regime with convergence guarantees for the iterates

Mathematical Programming - We present a trust-region steepest descent method for dynamic optimal control problems with binary-valued integrable control functions. Our method interprets the control...     

Mean Field Equation of Liouville Type with Singular Data: Topological Degree

We consider the following mean field equation: where M is a compact Riemann surface with volume 1, h* is a positive C¹ function on M, and ρ and α_j are constants satisfying α_j > ?1. In this paper, we derive the topological‐degree‐counting formula for noncritical values of ρ. We also give several applications of this formula, including existence of the curvature + 1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self‐gravitating strings. © 2015 Wiley Periodicals, Inc.     

ON AUGMENTED LAGRANGIAN METHODS FOR SADDLE-POINT LINEAR SYSTEMS WITH SINGULAR OR SEMIDEFINITE （1, 1） BLOCKS

An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite （1, 1） blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems.     

基于尺度赋权方差的有偏总体EWMA均值控制图

基于尺度赋权方差法给出总体分布有偏并假定总体分布未知情况下休哈特均值控制图的控制限.基于此研究结果,将其推广到EWMA均值控制图,给出总体分布有偏并假定总体分布未知情况下EWMA均值控制图的控制界限,并以质量特性值服从对数正态分布为例,给出EWMA均值控制图的控制效果的模拟分析.