A study of interval optimization problems

We study optimization problems with interval objective functions. We focus on set-type solution notions defined using the Kulisch–Miranker order between intervals. We obtain bounds for the asymptotic cones of level, colevel and solution sets that allow us to deduce coercivity properties and coercive existence results. Finally, we obtain various noncoercive existence results. Our results are easy to check since they are given in terms of the asymptotic cone of the constraint set and the asymptotic functions of the end point functions. This work extends, unifies and sheds new light on the theory of these problems.

[1]  Johannes Jahn,et al.  New Order Relations in Set Optimization , 2011, J. Optim. Theory Appl..

[2]  Giuseppe Buttazzo,et al.  Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization (Mps-Siam Series on Optimization 6) , 2005 .

[3]  Daishi Kuroiwa.  Existence theorems of set optimization with set-valued maps , 2003 .

[4]  Xiaoqi Yang,et al.  Theorems of the Alternative and Optimization with Set-Valued Maps , 2000 .

[5]  Luciano Stefanini,et al.  A comparison index for interval ordering based on generalized Hukuhara difference , 2012, Soft Computing.

[6]  Nicolas Hadjisavvas,et al.  An Optimal Alternative Theorem and Applications to Mathematical Programming , 2007, J. Glob. Optim..

[7]  Willard L. Miranker,et al.  Computer arithmetic in theory and practice , 1981, Computer science and applied mathematics.

[8]  Hsien-Chung Wu.  On interval-valued nonlinear programming problems , 2008 .

[9]  Ralph E. Steuer.  Algorithms for Linear Programming Problems with Interval Objective Function Coefficients , 1981, Math. Oper. Res..

[10]  Ajay Kumar Bhurjee,et al.  Multi-objective interval fractional programming problems : An approach for obtaining efficient solutions , 2015 .

[11]  H. Ishibuchi,et al.  Multiobjective programming in optimization of the interval objective function , 1990 .

[12]  E. Hernández,et al.  About Asymptotic Analysis and Set Optimization , 2019 .

[13]  Giorgio C. Buttazzo,et al.  Variational Analysis in Sobolev and BV Spaces - Applications to PDEs and Optimization, Second Edition , 2014, MPS-SIAM series on optimization.

[14]  Ronald T. Kneusel,et al.  Numbers and Computers , 2015, Springer International Publishing.

[15]  Fabián Flores-Bazán.  Ideal, weakly efficient solutions for vector optimization problems , 2002 .

[16]  Constantin Zalinescu,et al.  Set-valued Optimization - An Introduction with Applications , 2014, Vector Optimization.

[17]  Yurilev Chalco-Cano,et al.  Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative , 2013, Fuzzy Optim. Decis. Mak..

[18]  Asoke Kumar Bhunia,et al.  A Comparative Study of Different Order Relations of Intervals , 2012, Reliab. Comput..

[19]  M. Teboulle,et al.  Asymptotic cones and functions in optimization and variational inequalities , 2002 .

[20]  Hsien-Chung Wu,et al.  The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function , 2007, Eur. J. Oper. Res..

[21]  Yurilev Chalco-Cano,et al.  Optimality conditions for generalized differentiable interval-valued functions , 2015, Inf. Sci..

[22]  Kazufumi Ito,et al.  A Note on the Existence of Nonsmooth Nonconvex Optimization Problems , 2014, J. Optim. Theory Appl..

[23]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .