# Robust semi-infinite interval-valued optimization problem with uncertain inequality constraints

## Main Article Content

## Abstract

This paper focuses on a robust semi-infinite interval-valued optimization problem with uncertain inequality constraints (RSIIVP). By employing the concept of LU-optimal solution and Extended Mangasarian-Fromovitz Constraint Qualification (EMFCQ), necessary optimality conditions are established for (RSIIVP) and then sufficient optimality conditions for (RSIIVP) are derived, by using the tools of convexity. Moreover, a Wolfe type dual problem for (RSIIVP) is formulated and usual duality results are discussed between the primal (RSIIVP) and its dual (RSIWD) problem. The presented results are demonstrated by non-trivial examples.

## Article Details

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.

## References

[1] . I.Ahmad, A.Jayswal and J.Banerjee, On interval-valued optimization problems with generalized invex functions, J. Inequal. Appl. 2013 (1) (2013), 1–14. Google Scholar

[2] I.Ahmad, K.Kummari and S.Al-Homidan, Sufficiency and duality for interval-valued optimization problems with vanishing constraints using weak constraint qualifications, Internat. J. Anal. Appl. 18 (5) (2020), 784–798. Google Scholar

[3] H.Azimian, R.V.Patel, M.D.Naish and B.Kiaii, A semi-infinite programming approach to preoperative planning of robotic cardiac surgery under geometric uncertainty, IEEE J. Biomed. Health Inform. 17(1) (2012), 172–182. Google Scholar

[4] C.Bandi and D.Bertsimas, Tractable stochastic analysis in high dimensions via robust optimization, Math. Program. 134 (1) (2012), 23–70. Google Scholar

[5] A.Ben-Tal, L.E.Ghaoui and A.Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics. Princeton University Press, (2009). Google Scholar

[6] A.Ben-Tal and A.Nemirovski, Selected topics in robust convex optimization, Math. Program. 112 (1)(2008), 125–158. Google Scholar

[7] . D.Bertsimas, D.B.Brown and C.Caramanis, Theory and applications of robust optimization, SIAM Rev. 53 (3) (2011), 464–501. Google Scholar

[8] S¸.˙I.Birbil, J.B.G.Frenk, J.A.Gromicho and S.Zhang, The role of robust optimization in single-leg airline revenue management, Management Sci. 55 (1) (2009), 148–163. Google Scholar

[9] J.F.Bonnans and A.Shapiro, Perturbation Analysis of Optimization Problems, New York and Springer, (2013). Google Scholar

[10] A.K.Bhurjee and G.Panda, Efficient solution of interval optimization problem, Math. Methods Oper. Res. 76 (3) (2012), 273–288. Google Scholar

[11] C.Caramanis, S.Mannor and H.Xu, 14 Robust optimization in machine learning, In: S.Sra, S.Nowozin and S.Wright (editors). Optimization for Machine Learning, MIT Press, (2012), 369–402. Google Scholar

[12] A.Charnes, W.W.Cooper and K.Kortanek, A duality theory for convex programs with convex constraints, Bull. Amer. Math. Soc. 68 (6) (1962), 605–608. Google Scholar

[13] S.L.Chen, The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds, Optimization (2020), doi: 10.1080/02331934.2020.1810248 Google Scholar

[14] B.A.Dar, A.Jayswal and D.Singh, Optimality, duality and saddle point analysis for intervalvalued non-differentiable multi-objective fractional programming problems, Optimization (2020), doi: 10.1080/02331934.2020.1819276 Google Scholar

[15] V.Gabrel, C.A.Murat and A.Thiele, Recent advances in robust optimization: An overview, European J. Oper. Res. 235 (3) (2014), 471–483. Google Scholar

[16] M.Goerigk and A.Sch¨obel, Algorithm engineering in robust optimization, In: L.Kliemann and P.Sanders (editors). Algorithm Engineering, Lect. Notes Comput. Sci. Springer, Cham, (2016), 245– 279. Google Scholar

[17] R.Hettich and K.O.Kortanek, Semi-infinite programming: Theory, methods and applications, SIAM Rev. 35 (3) (1993), 380–429. Google Scholar

[18] A.Hussain, V.H.Bui and H.M.Kim, Robust optimization-based scheduling of multi-microgrids considering uncertainties, Energies 9 (4) (2016), 278. Google Scholar

[19] A.Jayswal, J.Banerjee and R.Verma, Some relations between interval-valued optimization and variational-like inequality problems, Comm. Appl. Nonlinear Anal. 20 (4) (2013), 47–56. Google Scholar

[20] V.Jeyakumar, G.M.Lee and G.Li, Characterizing robust solution sets of convex programs under data uncertainty, J. Optim. Theory Appl. 164 (2) (2015), 407–435. Google Scholar

[21] K.O.Kortanek and V.G.Medvedev, Semi-infinite programming and applications in finance, In: C.A.Floudas, P.M. Pardalos(editors). Encyclopaedia of Optimization, Boston and Springer, MA, (2008). Google Scholar

[22] K.Kummari and I.Ahmad, Sufficient optimality conditions and duality for non-smooth interval-valued optimization problems via L-invex-infine functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (1) (2020), 45–54. Google Scholar

[23] P.Kumar, B.Sharma and J.Dagar, Multiobjective semi-infinite variational problem and generalized invexity, Opsearch 54 (3) (2017), 580–597. Google Scholar

[24] P.Kumar and J.Dagar, Optimality and duality for multi-objective semi-infinite variational problem using higher-order B-type I functions, J. Oper. Res. Soc. China 9 (2) (2021), 375–393. Google Scholar

[25] K.K.Lai, S.Y.Wang, J.P.Xu, S.S.Zhu and Y.Fang, A class of linear interval programming problems and its application to portfolio selection, IEEE Trans. Fuzzy Syst. 10 (6) (2002), 698–704. Google Scholar

[26] J.H.Lee and G.M.Lee, On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems, Ann. Oper. Res. 269 (1) (2018), 419–438. Google Scholar

[27] J.Lin, M.Liu, J.Hao and S.Jiang, A multi-objective optimization approach for integrated production planning under interval uncertainties in the steel industry, Comput. Oper. Res. 72 (2016), 189–203. Google Scholar

[28] M.S.Pishvaee, M.Rabbani and S.A.Torabi, A robust optimization approach to closed-loop supply chain network design under uncertainty, Appl. Math. Model. 35 (2) (2011), 637–649. Google Scholar

[29] E.Polak, Semi-infinite optimization in engineering design, In: A.V.Fiacco and K.O.Kortanek (editors). Semi-Infinite Programming and Applications, Lect. Notes Econ. Math. Syst. Berlin and Springer, 215 (1983). Google Scholar

[30] E.W.Sachs, Semi-infinite programming in control, In: R.Reemtsen, J.J.R¨uckmann (editors). Semi-Infinite Programming. Nonconvex Optimization and its Applications, Boston and Springer, MA, 25 (1998), 389–411. Google Scholar

[31] Y.Shi, T.Boudouh and O.Grunder, A robust optimization for a home health care routing and scheduling problem with consideration of uncertain travel and service times, Transp. Res. E Logist. Transp. Rev. 128 (2019), 52–95. Google Scholar

[32] A.A.Shaikh, L.E.C´ardenas-Barr´on and S.Tiwari, A two-warehouse inventory model for noninstantaneous deteriorating items with interval-valued inventory costs and stock-dependent demand under inflationary conditions, Neural. Comput. Appl. 31 (6) (2019), 1931–1948. Google Scholar

[33] D.Singh, B.A.Dar and A.Goyal, KKT optimality conditions for interval-valued optimization problems, J. Nonlinear Anal. Optim. 5 (2) (2014), 91–103. Google Scholar

[34] A.C.Tolga, I.B.Parlak and O.Castillo, Finite-interval-valued Type-2 Gaussian fuzzy numbers applied to fuzzy TODIM in a healthcare problem, Eng. Appl. Artif. Intell. 87 (2020), 103352. Google Scholar

[35] L.T.Tung, Karush-Kuhn-Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions, J. Appl. Math. Comput. 62 (2020), 67–91. Google Scholar

[36] F.G.V´azquez and J.J.R¨uckmann, Semi-infinite programming: Properties and applications to economics, In: J.Leskow, L.F.Punzo and M.P.Anyul (editors). New Tools of Economic Dynamics, Lect. NotesEcon. Math. Syst. Springer, 551 (2005). Google Scholar

[37] A.I.F.Vaz and E.C.Ferreira, Air pollution control with semi-infinite programming, Appl. Math. Model. 33 (4) (2009), 1957–1969. Google Scholar

[38] B.Zhang, Q.Li, L.Wang and W.Feng, Robust optimization for energy transactions in multi-microgrids under uncertainty, Appl. Energy 217 (2018), 346–360. Google Scholar

[39] J.Zhang, Q.Zheng, C.Zhou, X.Ma and L.Li, On interval-valued pseudo-linear functions and intervalvalued pseudo-linear optimization problems, J. Funct. Space 2015 (2015), Article ID 610848 Google Scholar