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Some Duality Results in Multiple Objective Linear and Nonlinear Programming and a Nonmonotone Quasi-Newton Algorithm for Unconstrained Multiple Objective Optimization

Salehi Sadaghiani, Farnaz | 2017

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 50196 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezam Oddin
  7. Abstract:
  8. Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established.Here, we use these results to establish some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary slackness conditions exist. Furthermore,we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary slackness conditions for Isermann’s dual problem.Recently, Gupta, Kailey and Sharma utilized higher order ( Ϝ,α , ρ , d )-convexity.Here, we develop some duality results such as weak, strong and strict converse duality theorems between multiple objective nonlinear programming problems with inequality and equality constraints and their higher order Mangasarian type duals,under higher order ( Ϝ,α , ρ , d )-convexity assumptions. We also establish conditions for primal and dual feasible points to be efficient, for unboundedness of dual objective functions on the dual feasible region, and for primal problems not having efficient solutions. Then, we apply the obtained results to the special multiple objective nonlinear problem with bounded variables and state the corresponding special specifications. For solving unconstrained multiple objective programming problems, we propose and analyze a nonmonotone quasi-Newton algorithm for unconstrained multiple objective optimization. In our method, we allow for the decrease of convex combinations of recent function values. We prove global convergence and local superlinear convergence under reasonable assumptions. We implement our scheme in the context of BFGS quasi-Newton method for solving unconstrained multiple objective optimization problems. Our numerical results show that the nonmonotone quasi-Newton algorithm uses fewer function evaluations and less computing times than the monotone quasi-Newton algorithm
  9. Keywords:
  10. Efficient Solvent ; Multi Objective Programming ; Duality Theory ; Nonmonotone Line Search Algorithm ; Nonmonotone Quasi-Newton Method ; Strictly Feasible Points ; Strictly Complementary Slackness Condition

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