108 research outputs found
An elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions in nonlinear programming
In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints.The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization ofFarkas lemma and the Bolzano-Weierstrass property for compact sets.Fritz-John conditions;Karush-Kuhn-Tucker conditions;nonlinear programming
An elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions in nonlinear programming
In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints.The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of
Farkas lemma and the Bolzano-Weierstrass property for compact sets
On the multiplier rules
We establish new results of first-order necessary conditions of optimality
for finite-dimensional problems with inequality constraints and for problems
with equality and inequality constraints, in the form of John's theorem and in
the form of Karush-Kuhn-Tucker's theorem. In comparison with existing results
we weaken assumptions of continuity and of differentiability.Comment: 9 page
On representations of the feasible set in convex optimization
We consider the convex optimization problem where is convex, the feasible set K is convex and Slater's
condition holds, but the functions are not necessarily convex. We show
that for any representation of K that satisfies a mild nondegeneracy
assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely
every KKT point is a minimizer. That is, the KKT optimality conditions are
necessary and sufficient as in convex programming where one assumes that the
are convex. So in convex optimization, and as far as one is concerned
with KKT points, what really matters is the geometry of K and not so much its
representation.Comment: to appear in Optimization Letter
Optimality conditions and duality for nondifferentiable multiobjective programming problems involving d-r-type I functions
AbstractIn this paper, new classes of nondifferentiable functions constituting multiobjective programming problems are introduced. Namely, the classes of d-r-type I objective and constraint functions and, moreover, the various classes of generalized d-r-type I objective and constraint functions are defined for directionally differentiable multiobjective programming problems. Sufficient optimality conditions and various Mond–Weir duality results are proved for nondifferentiable multiobjective programming problems involving functions of such type. Finally, it is showed that the introduced d-r-type I notion with r≠0 is not a sufficient condition for Wolfe weak duality to hold. These results are illustrated in the paper by suitable examples
Symbolic approaches and artificial intelligence algorithms for solving multi-objective optimisation problems
Problems that have more than one objective function are of great importance in engineering sciences and many other disciplines. This class of problems are known as multi-objective optimisation problems (or multicriteria). The difficulty here lies in the conflict between the various objective functions. Due to this conflict, one cannot find a single ideal solution which simultaneously satisfies all the objectives. But instead one can find the set of Pareto-optimal solutions (Pareto-optimal set) and consequently the Pareto-optimal front is established. Finding these solutions plays an important role in multi-objective optimisation problems and mathematically the problem is considered to be solved when the Pareto-optimal set, i.e. the set of all compromise solutions is found. The Pareto-optimal set may contain information that can help the designer make a decision and thus arrive at better trade-off solutions. The aim of this research is to develop new multi-objective optimisation symbolic algorithms capable of detecting relationship(s) among decision variables that can be used for constructing the analytical formula of Pareto-optimal front based on the extension of the current optimality conditions. A literature survey of theoretical and evolutionary computation techniques for handling multiple objectives, constraints and variable interaction highlights a lack of techniques to handle variable interaction. This research, therefore, focuses on the development of techniques for detecting the relationships between the decision variables (variable interaction) in the presence of multiple objectives and constraints. It attempts to fill the gap in this research by formally extending the theoretical results (optimality conditions). The research then proposes first-order multi-objective symbolic algorithm or MOSA-I and second-order multi-objective symbolic algorithm or MOSA-II that are capable of detecting the variable interaction. The performance of these algorithms is analysed and compared to a current state-of-the-art optimisation algorithm using popular test problems. The performance of the MOSA-II algorithm is finally validated using three appropriately chosen problems from literature. In this way, this research proposes a fully tested and validated methodology for dealing with multi-objective optimisation problems. In conclusion, this research proposes two new symbolic algorithms that are used for identifying the variable interaction responsible for constructing Pareto-optimal front among objectives in multi-objective optimisation problems. This is completed based on a development and relaxation of the first and second-order optimality conditions of Karush-Kuhn-Tucker.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
An Elementary Proof of the Fritz-John and Karush-Kuhn-Tucker Conditions in Nonlinear Programming
In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets.Fritz-John conditions;Karush-Kuhn-Tucker conditions;nonlinear programming
Pseudonormality and a language multiplier theory for constrained optimization
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (leaves 211-213).Lagrange multipliers are central to analytical and computational studies in linear and non-linear optimization and have applications in a wide variety of fields, including communication, networking, economics, and manufacturing. In the past, the main research in Lagrange multiplier theory has focused on developing general and easily verifiable conditions on the constraint set, called constraint qualifications, that guarantee the existence of Lagrange multipliers for the optimization problem of interest. In this thesis, we present a new development of Lagrange multiplier theory that significantly differs from the classical treatments. Our objective is to generalize, unify, and streamline the theory of constraint qualifications. As a starting point, we derive an enahanced set of necessary optimality conditions of the Fritz John-type, which are stronger than the classical Karush-Kuhn-Tucker conditions. They are also more general in that they apply even when there is a possibly nonconvex abstract set constraint, in addition to smooth equality and inequality constraints. These optimality conditions motivate the introduction of a new condition, called pseudonormality, which emerges as central within the taxonomy of significant characteristics of a constraint set. In particular, pseudonormality unifies and extends the major constraint qualifications. In addition, pseudonormality provides the connecting link between constraint qualifications and exact penalty functions. Our analysis also yields identification of different types of Lagrange multipliers. Under some convexity assumptions, we show that there exists a special Lagrange multiplier vector, called informative, which carries significant sensitivity information regarding the constraints that directly affect the optimal cost change.(cont.) In the second part of the thesis, we extend the theory to nonsmooth problems under convexity assumptions. We introduce another notion of multiplier, called geometric, that is not tied to a specific optimal solution and does not require differentiability of the cost and constraint functions. Using a line of development based on convex analysis, we develop Fritz John-type optimality conditions for problems that do not necessarily have optimal solutions. Through an extended notion of constraint pseudonormality, this development provides an alternative pathway to strong duality results of convex programming. We also introduce special geometric multipliers that carry sensitivity information and show their existence under very general conditions.by Asuman E. Ozdaglar.Ph.D
Duality in mathematical programming.
In this thesis entitled, “Duality in Mathematical Programming”, the emphasis is given on formulation and conceptualization of the concepts of second-order duality, second-order mixed duality, second-order symmetric duality in a variety of nondifferentiable nonlinear programming under suitable second-order convexity/second-order invexity and generalized second-order convexity / generalized second-order invexity. Throughout the thesis nondifferentiablity occurs due to square root function and support functions. A support function which is more general than square root of a positive definite quadratic form. This thesis also addresses second-order duality in variational problems under suitable second-order invexity/secondorder generalized invexity. The duality results obtained for the variational problems are shown to be a dynamic generalization for thesis of nonlinear programming problem.Digital copy of Thesis.University of Kashmir
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