The paper deals with vector constrained extremum problems. Lagrangian duality and convex optimization duration. In particular, a strong duality theorem is given, new concepts for saddle points in the setvalued framework are introduced and the relationship between primaldual solutions and saddle points is established. The maximization with respect to a cone of a setvalued function into possibly infinite dimensions is defined. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions. Duality and penalization in optimization via an augmented lagrangian function with applications 11 october 2008 journal of optimization theory and applications, vol. Lagrangian duality in setvalued optimization springerlink.
Lagrangian duality for preinvex setvalued functions. Select publications by professor jeya jeyakumar unsw research. If minimising the lagrangian over xhappens to be easy for our problem, then we know that maximising the resulting dual function over is easy. This paper constructs an alternative dual that is differentiable.
We can consider this field as the ultimate extension of optimization problems involving a set. Lagrangian duality for multiobjective programming problems. Strong lagrangian duality holds for the quadratic programming with a twosided quadratic constraint. Recently, lagrangian duality for set valued optimization problems using completelattice theory was studied in 5,8, 16.
She publishes extensively in nonsmoothconvexnonconvex optimization, variational inequalities, and set valued analysis. In this paper, by using a alternative theorem, we establish lagrangian conditions and duality results for setvalued vector optimization problems when the objective and constant are nearly conesubconvexlike multifunctions in the sense of weak minimizer. Chapters 3, 5, and 6 constitute a unified logical unit that deals with smooth optimization issues. Mathematics and computer science college of arts and. Lagrange multiplier approach to variational problems and. Lagrangian duality for scalar and vector optimization problems. Some relationships between the solutions of the primal and the dual problems are discussed. Editor shashi kant mishra banaras hindu university faculty of science dept. We establish some duality theorems and present several existence results of a lagrange multiplier and a lexicographic saddle point theorem. An introduction to continuous optimization for imaging acta. Huy n, 2008, global minimization of difference of quadratic and convex functions over box or binary constraints, optimization letters, vol.
A lagrangian type dual has been defined for a fractional programming problem involving preinvex setvalued functions and duality results are established. Lagrangian duality theory refers to a way to find a bound or solve an. Jeyakumar v, 2008, constraint qualifications characterizing lagrangian duality in convex optimization, journal of optimization theory and applications, vol. The material in this tutorial is excerpted from the recent book on convex optimization, by boyd and vandenberghe, who have made available a large amount of free course material and freely available software. This paper deals with a constraint multiobjective programming problem and its dual problem in the lexicographic order. Citeseerx lagrange duality of setvalued optimization. Subdifferential necessary conditions for extremal solutions to set valued optimization problems with equilibrium constraints, bao q. A very good book for this subject is convex optimization by boyd and vandenberghe. In the following sections, our solution concept for setvalued optimization. In contrast to previous approaches, setvalued versions for the known scalar formulas involving infimum and supremum are obtained. A duality theorem of the fenchelrockafellar type for setvalued optimization problems is presented along with a result for the conjugate of the sum of two setvalued functions and a chain rule. Spcom 2015 will take place soon after another australian mathematics meeting, anziam 2015, which will be held between 1 and 5 february 2015 in outrigger surfers. Regina burachik is an associate professor in optimization at university of south australia.
Theoretically, if the solution to the original optimization from above is. Based on this, several lagrange multiplier type as well as saddle point type necessary and sufficient conditions are obtained for the existence of weak minimizers in vector optimization of set valued maps. Pdf lagrange duality, stability and subdifferentials in. The strong duality is an inclusion between the set of infimum points of the original multiobjective constrained optimization problem and that of the nonlinear lagrangian dual problem. In this work, we provide a nontrivial extension of 9 to 3d slam.
A fenchelrockafellar duality theorem for setvalued optimization. Duality for setvalued vector optimization problems is an important class of duality theory. In section 2, we recall some basic definitions, the m thorder contingent set and the m thorder adjacent set. A scalarization concept is introduced that has a meaning in terms of internal prices of portfolios of reference instruments. We proposed a duality approach in a paper on inverse optimization with noisy data 6, though the prior formulation is not differentiable because of the use of lagrangian duals. Nonlinear lagrangian for multiobjective optimization and applications to duality and exact penalization.
Lagrangian duality for multiobjective programming problems in. Pdf lagrange duality in set optimization researchgate. Oct 18, 2005 doubleregularization proximal methods, with complementarity applications doubleregularization proximal methods, with complementarity applications silva, paulo. In mathematical optimization, the method of lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints i. Theconjugatedualityforsetvaluedvectoroptimization problems has been made in 8. South pacific continuous optimization meeting spcom 2015 amsi. A set optimization approach to utility maximization under transaction. Weak duality the column geometry of the primal and dual problems strong duality duality strategies illustration of lagrange duality in discrete optimization 2 the practical importance of duality duality arises in nonlinear and linear optimization models in a wide variety of settings. Lagrangian duality for dummies stanford computer science. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. In this paper, we show that the twosided quadratic constrained quadratic fractional programming, if well scaled, also has zero lagrangian duality gap.
Also, cone lower semicontinuities of setvalued maps are introduced, and existence theorems of solutions of such problems are established. The basic idea is to convert a constrained problem into a form such that the. In this paper, we study optimization problems where the objective function and the binding constraints are setvalued maps and the solutions are defined by means of setrelations among all the images sets kuroiwa, d. Effective heuristics for the facilities location problem.
Duality theory for optimization problems with intervalvalued. Examples of left, secondleft convex and right, secondright nonconvex sets in r2. Higherorder generalized adjacent derivative and applications to duality for setvalued optimization wang, q. Opmt 5701 optimization with constraints the lagrange. In contrast to previous approaches, setvalued versions for the known. Lagrangian function can be viewed as a function aggregated the original objective function plus the. The lexicographic cone of is defined as the set of all vectors whose first.
Since s and t are nonempty convex sets with empty intersection, according. Conjugate duality for setvalued optimization of benson. This paper is devoted to developing augmented lagrangian duality theory in vector optimization. The lagrangian duality for setvalued vector optimization problems was studied byliandchen6andsong7. A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The theory is illustrated with an application to optimal control of an ordinary differential equation ode.
Lagrangian duality and branchandbound algorithms for. Linear programming, lagrange multipliers, and duality. A farkasminkowski type theorem has been proved for these functions. Duallagrangian methods for constrained optimization. This method involves adding an extra variable to the problem called the lagrange multiplier, or we then set up the problem as follows. Lagrangian method constrained optimization youtube. If an interiorpoint x is a localglobal minimizer, then f. To dualspace theory of set valued optimization, bao q. Moreover, from a geometric point of view, it is shown that the new method has a long stepsize, and it guarantees that the distance from the next iterative point to the solution set has a. On augmented lagrangian duality in integer programming andrew eberhard, rmit university email.
In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak. Henig saddle points and duality of setvalued optimization problems in the setting of real linear spaces. Set optimization problems with objective setvalued maps are considered, and some criteria of solutions are defined. Abstract in this article, we construct a fenchel lagrangian. We introduce a new dual problem, establish some duality theorems and obtain. Duality for set valued vector optimization problems is an important class of duality theory. A fenchelrockafellar duality theorem for setvalued. Teboulle a probabilistic result for the maxcut problem on random graphs. C constraint qualification for inexact linear programs. Based on the completelattice approach, a new lagrangian duality theory for setvalued optimization problems is presented. Laurent alfandari globalized gammarobustness christina busing, andreas barmann, frauke liers robust optimization for nonlinear impact of data variation laurent alfandari, juan carlos espinoza garcia lagrangian duality for a class of twostage robust problems. I understand the concept of duality in convex optimization. In this paper, by using a alternative theorem, we establish lagrangian conditions and duality results for setvalued vector optimization problems when the objective and constant are nearly conesubconvexlike.
Learn vocabulary, terms, and more with flashcards, games, and other study tools. Education the hong kong polytechnic university polyu. Lagrange duality in set optimization acm digital library. Asymptotic lagrangian duality for nonsmooth optimization tuesday, july 3, 2018. The solution to the dual problem provides a lower bound to the solution of the primal minimization problem. Unravelling nondifferentiable manifold problems based on. Lagrangian duality in setvalued optimization request pdf. Lagrangian duality for vector optimization problem with setvalued mappings peng, jianwen and long, xian, taiwanese journal of mathematics, 20 ekelands variational principle for setvalued maps with applications to vector optimization in uniform spaces ansari, qamrul, eshghinezhad, somayeh, and fakhar. A unified augmented lagrangian approach to duality and exact. Based on the completelattice approach, a new lagrangian type duality theory for set valued optimization problems is presented. L a duality theory for convex programming with setinclusive constraints. The fourth south pacific continuous optimization meeting spcom 2015 is hosted by the university of south australia and held in adelaide, australia, between 8 and 12 february 2015.
Here is the infinite dimensional version of the lagrange multiplier theorem for convex problems with inequality constraints. Professor stephen boyd, of the stanford university electrical engineering department, lectures on duality in the realm of electrical engineering and how it is utilized in convex optimization for. However, i am not able to understand how we can use it to solve problems. The focus of this research area is on the development of optimization theory, methods, and software and in particular the development and analysis of methods and their implementation in highquality software. Lagrangian multipliers, saddle points, and duality in vector. Secondly, under the assumption of the generalized cone subconvexlikeness of setvalued maps, the relationship between the. Atri rudra in this lecture we will cover some basic stuff on optimization. Lagrange duality, stability and subdifferentials in vector optimization article pdf available in optimization 623 november 2012 with 81 reads how we measure reads. Lagrangian duality for vector optimization problems with setvalued mappings xianjun long and jianwen peng abstract. The lagrangian dual problem is obtained by forming the lagrangian of a. Set valued optimization arises quite naturally in the topic of duality for vector optimization see e. We propose an improved twostep extragradient algorithm for pseudomonotone generalized variational inequalities. First order methods beyond convexity and lipschitz gradient continuity with applications to quadratic inverse problems. Duality optimization in mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem.
Teboulle lagrangian multiplier methods in convex programming. In contrast to previous approaches, set valued versions for the known scalar formulas involving infimum and supremum are obtained. The topics of duality and interior point algorithms will be our focus, along with simple examples. Jun 14, 2007 in this paper, we study optimization problems where the objective function and the binding constraints are setvalued maps and the solutions are defined by means of setrelations among all the images sets kuroiwa, d. This article addresses a new approach to duality in setvalued optimization, by means of solutions defined with the help of the nonempty quasi interior of a convex cone, employing a settype. This paper is concerned with a lagrange duality theory for convex setvalued optimization with a setvalued constraint. Is it just used to get a quick lower bound in case of weak duality. Higherorder mondweir duality for setvalued optimization. A dualbased algorithm for solving lexicographic multiple objective programs.
The lagrangian involves a setvalued analog of continuous linear. Optimization is a fundamental tool in many areas of science, engineering, economics, and finance, including its use in machine learning. However in general the optimal values of the primal and dual problems need not. In mathematical optimization theory, duality or the duality principle is the principle that. The intervalvalued lagrangian function and intervalvalued. We propose and analyse a regularization method for parameter identification problems modeled by illposed nonlinear operator equations, where the parameter to be identified is a p. Journal of optimization theory and applications archive. Let kkbe any norm on rdsuch as the euclidean norm kk 2, and let x 0 2rd, r0.
Lagrangian duality in setvalued optimization article in journal of optimization theory and applications 41. Here gy is the dual function of fx,y duality gap and strong duality. The purpose of this paper is to present some results for lagrange fenchel dualities, via a perturbation approach and a conjugate mapping for the setvalued optimization which is under the assumption of benson efficiency. Jerome bolte, shoham sabach, marc teboulle, yakov vaisbourd.
For a randomly generated 3,000 by 3,000 pseudo correlation matrix the code is insensitive to input data, the code needs 24 seconds to reach a solution with the relative duality gap less than 1. It requires two projections at each iteration and allows one to take different stepsize rules. Furthermore, lagrange duality for setvalued problems based on vector approach is shown. Institute for mathematics, martinlutheruniversity hallewittenberg, halle. The main tool is a setvalued lagrangian, to which recent duality results for set valued functions from 10 are applied compare also the survey 8. Lagrangian duality, borrowing the corresponding theory from the optimization community. Approximate fenchellagrangian duality for constrained set. Ozdaglar a geometric framework for nonconvex optimization duality using augmented lagrangian functions journal of global optimization 40 4 545573, 2008. Strong duality we say that strong duality holds if the optimal duality gap is zero, i. Doubleregularization proximal methods, with complementarity. Exact penalization is established via a generalized calmnesstype condition. We establish weak duality, strong duality and converse duality results for optimization problems with setvalued mappings. Primal problems can be directly solved using newtons method or some other method.
By using the concepts of the supremum and infimum of a set and conjugate duality of a set valued map on the basic of weak efficiency, we establish the interchange rules for a set valued map, and propose an augmented lagrangian function for a vector optimization problem with set valued data. Department of mathematical sciences, yeshiva university new york, new york, usa 10033. Duality gives us an option of trying to solve our original potentially nonconvex constrained optimisation problem in another way. Duality assertions are very important in optimization researches from the. We introduce a new dual problem, establish some duality theorems and obtain a. Approximate lagrangian duality and saddle point optimality in set. We introduce a new dual problem, establish some duality. Regina burachik home page, university of south australia. Henig saddle point of the lagrangian setvalued map is obtained. If the constrained optimization problem is wellposed that is, has a finite and achievable minimum, the resulting game has a finite value which is equal to the value of the lagrangian at its saddle point. Extending the approach of jouini, meddeb, and touzi finance stoch.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Based on the completelattice approach, a new lagrangian type duality theory for setvalued optimization problems is presented. The lagrangian duality for set valued vector optimization problems was studied byliandchen6andsong7. Microeconomics i maximize utility with lagrange i example 2. This massive and very wellwritten book will probably be for many years the primary reference in setvalued optimization, the area of optimization where the objective andor constraint functions of the considered problems are setvalued maps. A good dual solution can be used to bound the values of primal solutions, and so can be used to actually identify when a primal solution is nearoptimal. An improved twostep method for generalized variational. Duality in vector optimization with infimum and supremum. The notions of conjugate maps, subdifferential and a perturbation approach used for deriving these duality assertions are given. Let cbe the corresponding norm ball of radius rcentered at x 0. In this section, we consider a possibly nonconvex optimization problem where the functions we denote by the domain of the problem which is the intersection of the domains of all the functions involved, and by its feasible set we will refer to the above as the primal problem, and to the decision variable in that problem, as the primal. Lagrangian duality and related multiplier methods for variational inequalities. Meanwhile, we study the relations between the lexicographic saddle point and the lexicographic solution to a. Duality for setvalued measures of risk siam journal on.
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