(635e) Jointly Robust Optimization for Multiple Uncertain Constraints

Optimization under uncertainty is important for many decision making problems. Among the various approaches dealing with uncertainty in optimization problem, the advantages of robust optimization include the computational tractability as well as the absence of the distribution information requirement. It deals with the uncertainty in the parameters by introducing an uncertainty set which covers part of the whole uncertainty space. The target is to select the best solution that is feasible for any realizations of the uncertain parameters in the uncertainty set [1]. In the open literature, most of the robust optimization methods assumed that the uncertainties in the parameters are independent [2,3]. But in practice, correlation may exist among different uncertain parameters. As a result, the traditional robust optimization methods that ignore the correlation may lead to a solution that is too conservative. The major issue of robust optimization is the design of the uncertainty set. Different types of uncertainty set lead to different robust counterpart formulations [2]. Recently, robust optimization under correlated uncertainties within a single constraint has been studied by incorporating the covariance matrix in the uncertainty set design, and corresponding robust counterpart formulations were derived [4].

In this work, a novel jointly robust formulation for optimization under independent or correlated uncertainties across multiple constraints is proposed. First, for a set of inequality constraints with uncertainty y₀ⁱ + (yⁱ)^Tζâ?¤0 (where, y₀ⁱand yⁱ are variables and ζ is uncertain parameter, iis the index for different constraints), it is converted to a single constraint by introducing a maximizing operator. Second, the selection of the largest left-hand-side is realized by introducing binary variables. Next, the mixed-integer constrained problem is equivalently transferred to a linear programming problem by relaxed the binary variables as continuous variables. Third, by incorporating the covariance matrix of the uncertainties in the uncertainty set as a general form, the robust counterpart can be formulated based on conic duality. With specified type of uncertainty set, the final robust counterpart formulations can be obtained. Lastly, the proposed jointly robust optimization formulation can be improved by introducing adjustable coefficient to each individual constraints, which can lead to a robust solution of better quality. Numerical examples and applications in process operations problems are studied to illustrate the proposed jointly robust optimization framework for uncertainty in multiple constraints.

Reference:

[1] Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations research letters, 25(1), 1-13.

[2] Li, Z., Ding, R., & Floudas, C. A. (2011). A comparative theoretical and computational study on robust counterpart optimization: I. Robust linear optimization and robust mixed integer linear optimization. Industrial & engineering chemistry research, 50(18), 10567-10603.

[3] Li, Z., Tang, Q., & Floudas, C. A. (2012). A comparative theoretical and computational study on robust counterpart optimization: II. Probabilistic guarantees on constraint satisfaction. Industrial & engineering chemistry research, 51(19), 6769-6788.

[4] Yuan, Y., Li, Z., & Huang, B. (2016). Robust optimization under correlated uncertainty: Formulations and computational study. Computers & Chemical Engineering, 85, 58-71.