Abstract
We investigate robust optimization problems defined for maximizing convex functions. While the problems arise in situations which are naturally modeled as minimization of concave functions, they also arise when a decision maker takes an optimistic approach to making decisions with convex functions. For finite uncertainty set, we develop a geometric branch-and-bound algorithmic approach to solve this problem. The geometric branch-and-bound algorithm performs sequential piecewise-linear approximations of the convex objective, and solves linear programs to determine lower and upper bounds at each node. Finite convergence of the algorithm to an \(\epsilon -\)optimal solution is proved. Numerical results are used to discuss the performance of the developed algorithm. The algorithm developed in this paper can be used as an oracle in the cutting surface method for solving robust optimization problems with compact ambiguity sets.
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References
Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable NLPs-I. Theoretical advances, Comput. Chem. Eng. 22(9), 1137–1158 (1998)
Adjiman, C..S., Dallwig, S., Floudas, C..A., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable NLPs-II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159–1179 (1998)
Bansal, M., Huang, K., Mehrotra, S.: Decomposition algorithms for two-stage distributionally robust mixed binary programs. SIAM J. Optim. 28, 2360–2383 (2017)
Bansal, M., Zhang, Y.: Two-Stage Stochastic and Distributionally Robust \(p\)-Order Conic Mixed Integer Programs (2018). http://www.optimization-online.org/DB_FILE/2018/05/6630.pdf
Ben-Tal, A., EI Ghaoui, L., Nemirovski, A.: Robust Optimization, Princeton Series in Applied Mathematics, Princeton University Press (2009)
Ben-Tal, A., EL Ghaoui, L., Nemirovski, A.: Robustness, Handbook of Semidefinite Programming. In: Saigal, R., Vandenberghe, L., Wolkowicz, H. (eds.), pp. 139–162. Kluwer Academic, Dordrecht (2000)
Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998)
Ben-Tal, A., Nemirovski, A.: Robust solutions to uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)
Benson, H.P.: Fractional programming with convex quadratic forms and functions. Eur. J. Oper. Res. 173, 351–369 (2006)
Benson, H.P.: A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem. Eur. J. Oper. Res. 182(2), 597–611 (2007)
Benson, H.P.: Solving sum of ratios fractional programs via concave minimization. J. Optim. Theory Appl. 135, 1–17 (2007)
Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15, 780–804 (2005)
Bonami, P., Lee, J., Leyffer, S., Wächter, A.: Novel bound contraction procedure for global optimization of bilinear minlp problems with applications to water management problems. Comput. Chem. Eng. 35, 446–455 (2011)
Chen, C., Atamtürk, A., Oren, S.S.: A spatial branch-and-cut method for nonconvex qcqp with bounded complex variables. Math. Program. 165, 549–577 (2017)
Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming, Graduate Texts in Mathematics. Springer (2014)
Faria, D.C., Bagajewicz, M.J.: Novel bound contraction procedure for global optimization of bilinear minlp problems with applications to water management problems. Comput. Chem. Eng. 35, 446–455 (2011)
Floudas, C.A.: Deterministic Global Optimization: Theory, Algorithms and Applications, 1 edn., Springer (2000)
Gerard, D., Köppe, M., Louveaux, Q.: Guided dive for the spatial branch-and-bound. J. Glob. Optim. 68, 685–711 (2017)
Henrion, D., Lasserre, J.B.: Gloptipoly: global optimization over polynomials with MATLAB and SeDuMi. ACM Trans. Math. Softw. 29, 165–194 (2003)
Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35(3), 380–429 (1993)
Jiao, H., Guo, Y., Shen, P.: Global optimization of generalized linear fractional programming with nonlinear constraints. Appl. Math. Comput. 183, 717–728 (2006)
Kirst, P., Stein, O., Steuermann, P.: Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints. TOP 23, 591–616 (2015)
Leyffer, S.: Integrating sqp and branch-and-bound for mixed integer nonlinear programming. Comput. Optim. Appl. 14, 295–309 (2001)
Li, Y., Shu, J., Song, M., Zhang, J., Zheng, H.: Multisourcing supply network design: two-stage chance-constrained model, tractable approximations, and computational results. INFORMS J. Comput. 29(2), 287–300 (2017)
Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. Ser. B 103, 251–282 (2005)
Luo, F.Q.: A distributionally-Robust Service Center Location Problem with Decision Dependent Demand Induced from a Maximum Attraction Principle (2020). https://arxiv.org/pdf/2011.12514.pdf
Luo, F.Q., Mehrotra, S.: A Decomposition Method for Distributionally-robust Two-stage Stochastic Mixed-integer Conic Programs (2019). https://arxiv.org/pdf/1911.08713.pdf
Luo, F.Q., Mehrotra, S.: Robust Maximization of Piecewise-linear Convex Functions Using Mixed Binary Linear Programming Reformulation. Tech. report, Northwestern University, Department of Industrial Engineering and Management Science (2019)
Pozo, C., Guillén-Gosálbez, G., Sorribas, A., Jiménez, L.: A spatial branch-and-bound framework for the global optimization of kinetic models of metabolic networks. Ind. Eng. Chem. Res. 50(9), 5225–5238 (2010)
Pronzato, L., Müller, W.G.: Design of computer experiments: space filling and beyond. Stat. Comput. 22(3), 681–701 (2012)
Rudin, W.: Principles of Mathematical Analysis, Example Product Manufacturer (2013)
Ryoo, H.S., Sahinidis, N.V.: Global optimization of non convex NLPs and MINLPs with applications in process design. Comput. Chem. Eng. 19, 551–566 (1995)
Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8, 107–138 (1996)
Shectman, J.P., Sahinidis, N.V.: A finite algorithm for global minimization of separable concave programs. J. Glob. Optim. 12, 1–36 (1998)
Shen, P.P., Yuan, G.X.: Global optimization for the sum of generalized polynomial fractional functions. Math. Methods Oper. Res. 65(3), 445–459 (2007)
Shen, Z.M., Coullard, C., Daskin, M.S.: A joint location-inventory model. Transp. Sci. 37(1), 40–55 (2003)
Sherali, H.D.: Global optimization of nonconvex polynomial programming problems having rational exponents. J. Glob. Optim. 12(3), 267–283 (1998)
Sherali, H.D., Wang, H.: Global optimization of non convex factorable programming problems. Math. Program. Ser. A 89, 459–478 (2001)
Smith, E.M.B., Pantelides, C.C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimization of nonconvex minlps. Comput. Chem. Eng. 23, 457–478 (1999)
Smith, E.M.B., Pantelides, C.C.: Global optimization of general process models. In: Grossmann, I.E. (ed.) Global Optimization in Engineering Design, pp. 355–386. Kluwer Academic, Boston (1996)
Stein, O., Kirst, P., Steuermann, P.: An Enhanced Spatial Branch-and-Bound Method in Global Optimization with Nonconvex Constraints (2013). http://www.optimization-online.org/DB_FILE//2013/04/3810.pdf
Tawarmalani, M.: Mixed Integer Nonlinear Programs: Theory, Algorithms and Applications. University of Illinois, Urbana-Champaign (2001) (Ph.D. thesis)
Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. Ser. A 99, 563–591 (2004)
Thakur, L.: Domain contraction in nonlinear programming: minimizing a quadratic concave objective over a polyhedron. Math. Oper. Res. 16, 390–407 (1991)
Trutman, P.: Polynomial Optimization Problem Solver (2017). https://github.com/PavelTrutman/polyopt
Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)
Wang, S., Li, J., Mehrotra, S.: A Solution Approach to Distributionally Robust Chance-constrained Assignment Problems (2019). http://www.optimization-online.org/DB_FILE/2019/05/7207.pdf
Wang, Y., Shen, P., Zhian, L.: A branch-and-bound algorithm to globally solve the sum of several linear ratios. Appl. Math. Comput. 168, 89–101 (2005)
Wang, Y.J., Liang, Z.: A deterministic global optimization algorithm for generalized geometric programming. Appl. Math. Comput. 168, 722–737 (2005)
Wittek, P.: Algorithm 950: Ncpol2sdpa—sparse semidefinite programming relaxations for polynomial optimization problems of non-commuting variables. ACM Trans. Math. Softw. 41(21) (2003)
Xie, W.: On distributionally robust chance constrained programs with wasserstein distance. Math. Program. 186(1), 115–155 (2019)
Xu, E.: A Python Connector to IPOPT (2014–2018). https://github.com/xuy/pyipopt
Zamora, J.M., Grossmann, I.E.: Continuous global optimization of structured process systems models. Comput. Chem. Eng. 22(12), 1749–1770 (1998)
Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms. J. Glob. Optim. 14, 217–249 (1999)
Zou, Y., Chakrabarty, K.: Sensor deployment and target localization in distributed sensor networks. ACM Trans. Embed. Comput. Syst. 3(1), 61–91 (2004)
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The research of this paper is supported by the National Science Foundation Grant CMMI-1361942. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Luo, F., Mehrotra, S. A geometric branch and bound method for robust maximization of convex functions. J Glob Optim 81, 835–859 (2021). https://doi.org/10.1007/s10898-021-01038-7
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DOI: https://doi.org/10.1007/s10898-021-01038-7