We study a variant of the pessimistic bi-level optimization problem, which comprises constraints that must be satisfied for any optimal solution of a subordinate (lower-level) optimization problem. We present conditions that guarantee the existence of optimal solutions in such a problem, and we characterize the computational complexity of various subclasses of the problem. We then focus on problem instances that may lack convexity, but that satisfy a certain independence property. We develop convergent approximations for these instances, and we derive an iterative solution scheme that is reminiscent of the discretization techniques used in semi-infinite programming. We also present a computational study that illustrates the numerical behavior of our algorithm on standard benchmark instances.

McCormick (Math Prog 10(1): 1976) provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F • f , where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. The generalization interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method. In addition to extending the framework, the new result provides a tool for the proof of relaxations of specific functions. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormick's result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. These cases can be extended to allow composition of a variety of functions for which relaxations have been proposed.

The algorithm proposed in Mitsos (Optimization 60(10-11):1291(Optimization 60(10-11): -1308(Optimization 60(10-11): , 2011 for the global optimization of semi-infinite programs is extended to the global optimization of generalized semi-infinite programs. No convexity or concavity assumptions are made. The algorithm employs convergent lower and upper bounds which are based on regular (in general nonconvex) nonlinear programs (NLP) solved by a (black-box) deterministic global NLP solver. The lower bounding procedure is based on a discretization of the lower-level host set; the set is populated with Slater points of the lower-level program that result in constraint violations of prior upper-level points visited by the lower bounding procedure. The purpose of the lower bounding procedure is only to generate a certificate of optimality; in trivial cases it can also generate a global solution point. The upper bounding procedure generates candidate optimal points; it is based on an approximation of the feasible set using a discrete restriction of the lower-level feasible set and a restriction of the right-hand side constraints (both lower and upper level). Under relatively mild assumptions, the algorithm is shown to converge finitely to a truly feasible point which is approximately optimal as established from the lower bound. Test cases from the literature are solved and the algorithm is shown to be computationally efficient.

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Multi-stage robust optimization problems, where the decision maker can dynamically react to consecutively observed realizations of the uncertain problem parameters, pose formidable theoretical and computational challenges. As a result, the existing solution approaches for this problem class typically determine suboptimal solutions under restrictive assumptions. In this paper, we propose a robust dual dynamic programming (RDDP) scheme for multi-stage robust optimization problems. The RDDP scheme takes advantage of the decomposable nature of these problems by bounding the costs arising in the future stages through lower and upper cost to-go functions. For problems with uncertain technology matrices and/or constraint righthand sides, our RDDP scheme determines an optimal solution in finite time. If also the objective function and/or the recourse matrices are uncertain, our method converges asymptotically (but deterministically) to an optimal solution. Our RDDP scheme does not require a relatively complete recourse, and it offers deterministic upper and lower bounds throughout the execution of the algorithm. We demonstrate the promising performance of our algorithm in stylized instances of inventory management and energy planning problems.immunized against all parameter realizations in the uncertainty set Ξ, which we assume to be stage-wise rectangular. The cost vectors q t , the technology matrices T t , the recourse matrices W t and the right-hand side vectors h t may depend affinely on ξ t . We assume that ξ 1 is deterministic, and hence x 1 is a here-and-now decision. Problem (1) readily accommodates constraints that link decisions of more than two time periods by augmenting x t appropriately. Note that we do not assume that problem (1) has a relatively complete recourse, that is, there may be partial solutions x 1 , . . . , x t satisfying the constraints up to time period t that cannot be extended to complete solutions x 1 , . . . , x T satisfying all constraints.The multi-stage robust optimization problem (1) is convex, but it involves infinitely many decision variables and constraints. In fact, problem (1) is NP-hard even for T = 2 time stages (Guslitser 2002), and polynomial-time solvable subclasses of problem (1) are both rare and restrictive (Anderson and Moore 1990, Bertsimas et al. 2010, Gounaris et al. 2013). As a result, any solution scheme for problem (1) has to trade off the competing goals of optimality and computational tractability. Tractable conservative approximations to problem (1) can be obtained by restricting the decisions x t to functions with pre-selected structure, which are called decision rules. Popular classes of decision rules include affine (Guslitser 2002, Kuhn et al. 2011), segregated affine (Chen et al. 2008, Chen and Zhang 2009, Goh and Sim 2010), piecewise affine (Georghiou et al. 2015) and polynomial as well as trigonometric functions (Bertsimas et al. 2011) of the parameters ξ t . Decision rule approximations typically scale polynomially with the size of the problem. However, t...