Duality and Exact Penalization for Vector Optimization via Augmented Lagrangian

In this paper, we introduce an augmented Lagrangian function for a multiobjective optimization problem with an extended vector-valued function. On the basis of this augmented Lagrangian, set-valued dual maps and dual optimization problems are constructed. Weak and strong duality results are obtained. Necessary and sufficient conditions for uniformly exact penalization and exact penalization are established. Finally, comparisons of saddle-point properties are made between a class of augmented Lagrangian functions and nonlinear Lagrangian functions for a constrained multiobjective optimization problem.     

Some Results about Duality and Exact Penalization

In this paper, we introduce the concept of the valley at 0 augmenting function and apply it to construct a class of valley at 0 augmented Lagrangian functions. We establish the existence of a path of optimal solutions generated by valley at 0 augmented Lagrangian problems and its convergence toward the optimal set of the original problem and obtain the zero duality gap property between the primal problem and the valley at 0 augmented Lagrangian dual problem. Moreover, we establish the exact penalization representation results in the framework of valley at 0 augmented Lagrangian.     

On Duality in Nonconvex Vector Optimization in Banach Spaces Using Augmented Lagrangians

This paper shows how the use of penalty functions in terms of projections on the constraint cones, which are orthogonal in the sense of Birkhoff, permits to establish augmented Lagrangians and to define a dual problem of a given nonconvex vector optimization problem. Then the weak duality always holds. Using the quadratic growth condition together with the inf-stability or a kind of Rockafellar's stability called stability of degree two, we derive strong duality results between the properly efficient solutions of the two problems. A strict converse duality result is proved under an additional convexity assumption, which is shown to be essential.     

Exact Lagrangians in plumbings

Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Spin Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A ₂-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.     

Exact Lagrangians in A_n-surface singularities

In this paper we classify Lagrangian spheres in $A_n$ -surface singularities up to Hamiltonian isotopy. Combining with a result of Ritter (Geom Funct Anal 20(3):779–816, 2010), this yields a complete classification of exact Lagrangians in $A_n$ -surface singularities. Our main new tool is the application of a technique which we call ball-swappings and its relative version.     

Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization

In this paper, we study a class of constrained scalar set-valued optimization problems, which includes scalar optimization problems with cone constraints as special cases. We introduce (local) calmness of order??? for this class of constrained scalar set-valued optimization problems. We show that the (local) calmness of order??? is equivalent to the existence of a (local) exact set-valued penalty map.     

Augmented Lagrangians which are quadratic in the multiplier

We present a class of new augmented Lagrangian functions with the essential property that each member is concave quadratic when viewed as a function of the multiplier. This leads to an improved duality theory and to a related class of exact penalty functions. In addition, a relationship between Newton steps for the classical Lagrangian and the new Lagrangians is established.This work was supported in part by ARO Grant No. DAAG29-77-G-0125.     

Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming

In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the $\alpha $ BB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In the present research, the algorithm mentioned above will be improved in several crucial aspects. On the one hand, feasibility of the problem will not be required. Possible infeasibility will be detected in finite time by the new algorithms and optimal infeasibility results will be proved. On the other hand, finite termination results that guarantee optimality and/or feasibility up to any required precision will be provided. An adaptive modification in which subproblem tolerances depend on current feasibility and complementarity will also be given. The adaptive algorithm allows the augmented Lagrangian subproblems to be solved without requiring unnecessary potentially high precisions in the intermediate steps of the method, which improves the overall efficiency. Experiments showing how the new algorithms and results are related to practical computations will be given.     

An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians

We consider a primal optimization problem in a reflexive Banach space and a duality scheme via generalized augmented Lagrangians. For solving the dual problem (in a Hilbert space), we introduce and analyze a new parameterized Inexact Modified Subgradient (IMSg) algorithm. The IMSg generates a primal-dual sequence, and we focus on two simple new choices of the stepsize. We prove that every weak accumulation point of the primal sequence is a primal solution and the dual sequence converges weakly to a dual solution, as long as the dual optimal set is nonempty. Moreover, we establish primal convergence even when the dual optimal set is empty. Our second choice of the stepsize gives rise to a variant of IMSg which has finite termination.     

非线性规划问题的精确增广Lagrange函数

对于一般的非线性规划给出一种精确增广Lagrange函数,并讨论其性质.无需假设严格互补条件成立,给出了原问题的局部极小点与增广Lagrange函数在原问题的变量空间上的局部极小的关系.进一步,在适当的假设条件下,建立了两者的全局最优解之间的关系.     

Variational Analysis on Local Sharp Minima via Exact Penalization

In this paper we study local sharp minima of the nonlinear programming problem via exact penalization. Utilizing generalized differentiation tools in variational analysis such as subderivatives and regular subdifferentials, we obtain some primal and dual characterizations for a penalty function associated with the nonlinear programming problem to have a local sharp minimum. These general results are then applied to the ? _p penalty function with 0 ≤ p ≤ 1. In particular, we present primal and dual equivalent conditions in terms of the original data of the nonlinear programming problem, which guarantee that the ? _p penalty function has a local sharp minimum with a finite penalty parameter in the case of \(p\in (\frac {1}{2}, 1]\) and \(p=\frac {1}{2}\) respectively. By assuming the Guignard constraint qualification (resp. the generalized Guignard constraint qualification), we also show that a local sharp minimum of the nonlinear programming problem can be an exact local sharp minimum of the ? _p penalty function with p ∈ [0, 1] (resp. \(p\in [0, \frac {1}{2}]\)). Finally, we give some formulas for calculating the smallest penalty parameter for a penalty function to have a local sharp minimum.     

Calmness and Exact Penalization in Vector Optimization with Cone Constraints

In this paper, a (local) calmness condition of order α is introduced for a general vector optimization problem with cone constraints in infinite dimensional spaces. It is shown that the (local) calmness is equivalent to the (local) exact penalization of a vector-valued penalty function for the constrained vector optimization problem. Several necessary and sufficient conditions for the local calmness of order α are established. Finally, it is shown that the local calmness of order 1 implies the existence of normal Lagrange multipliers. Presented at the 6th International Conference on Optimization: Techniques and Applications, Ballarat, Australia, December 9–11, 2004 This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.     

Calmness and Exact Penalization in Vector Optimization under Nonlinear Perturbations

In this paper, we study the relationship between calmness and exact penalization for vector optimization problems under nonlinear perturbations. Some sufficient conditions for the problem calmness are also derived.     

Numerical Solution of Viscoplastic Flow Problems by Augmented Lagrangians

This paper describes an application of augmented Lagrangiantechniques to the numerical solution of quasistatic flow problemsin incompressible viscoplasticity, focusing on cases where theinternal viscoplastic dissipation potential is not a differentiablefunction of the material deformation rate. The stresses of elasticorigin are neglected, and the variational formulation of theseproblems is approximated via low-order mixed finite elements,which reduces the original problems to the constrained minimizationof a convex, but possibly not differentiable functional. Convergenceresults are proved or recalled, both for the finite elementapproximation and for the augmented Lagrangian algorithm. Adetailed study of the local minimization problems which occurin the augmented Lagrangian decomposition of the above problemsis also presented, together with several numerical results.     

Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Equality Constraints

In this paper we introduce an augmented Lagrangian type algorithm for strictly convex quadratic programming problems with equality constraints. The new feature of the proposed algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm. Global convergence and boundedness of the penalty parameter are proved and an error estimate is given that does not have any term that accounts for the inexact solution of the auxiliary problems. Numerical experiments illustrate efficiency of the algorithm presented     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

Augmented Lagrangian Methods for $p$-Harmonic Flows with the Generalized Penalization Terms and Application to Image Processing

In this paper, we propose a generalized penalization technique and a convex constraint minimization approach for the $p$-harmonic flow problem following the ideas in [Kang & March, IEEE T. Image Process., 16 (2007), 2251-2261]. We use fast algorithms to solve the subproblems, such as the dual projection methods, primal-dual methods and augmented Lagrangian methods. With a special penalization term, some special algorithms are presented. Numerical experiments are given to demonstrate the performance of the proposed methods. We successfully show that our algorithms are effective and efficient due to two reasons: the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately (even 2 inner iterations of the subproblem are enough). It is also observed that better PSNR values are produced using the new algorithms.     

Deformations of Symplectic Cohomology and Exact Lagrangians in ALE Spaces

ALE spaces are the simply connected hyperkähler manifolds which at infinity look like ${\mathbb{C}^{2}/G}ALE spaces are the simply connected hyperk?hler manifolds which at infinity look like \mathbbC²/G{\mathbb{C}^{2}/G}, for any finite subgroup G ì SL₂(\mathbbC){G \subset SL_2(\mathbb{C})}. We prove that all exact Lagrangians inside ALE spaces must be spheres. The proof relies on showing the vanishing of a twisted version of symplectic cohomology.     

Augmented Lagrangian Theory,Duality and Decomposition Methods for Variational Inequality Problems

In this paper, we develop the augmented Lagrangian theory and duality theory for variational inequality problems. We propose also decomposition methods based on the augmented Lagrangian for solving complex variational inequality problems with coupling constraints.