Adjustable Robust Optimization Reformulations of Two-Stage Worst-Case Regret Minimization Problems

Although the stochastic optimization paradigm exploits probability theory to optimize the tradeoff between risk and returns, robust optimization has gained significant popularity by reducing computation requirements through the optimization of the worst-case scenario in a set. An appealing alternative to stochastic and robust optimization consists in optimizing decisions using the notion of regret. Although regret minimization models are generally perceived as leading to less conservative decisions than those produced by robust optimization, their numerical optimization is a real challenge in general. In “Adjustable Robust Optimization Reformulations of Two-Stage Worst-case Regret Minimization Problems,” M. Poursoltani and E. Delage show how to reduce a two-stage worst-case absolute/relative regret minimization problem to a two-stage robust optimization one. This opens the way for taking advantage of recent advanced approximate and exact solution schemes for these hard problems. Their experiments corroborate the high-quality performance of affine decision rules as a popular polynomial-time approximation scheme, from which, under mild conditions, one can even expect exact regret-averse decisions.

Optimal Operation of Isolated Micro-Grids-cluster Via Coalitional Energy Scheduling and Reserve Sharing

Microgrids (MG) cluster are isolated from the utility grid but they have the potential to achieve better techno-economic performance by using joint energy and reserve sharing among MGs. This paper proposes a techno-economic framework for the optimal operation of isolated MGs-cluster by scheduling cooperative energy sharing and real-time reserve sharing for ancillary services based on the cooperative game theory. In the day-ahead scheduling, a coalitional sharing scheme is formulated as an adjustable robust optimization (ARO) problem to optimally schedule the energy and reserves of distributed generators (DGs) and energy storage systems (ESSs), thereby responding to the uncertainties of photovoltaic systems, wind turbines, and loads. These uncertainties are the main reason for power system imbalance which is mitigated by regulating the frequency in real-time and a dynamic droop control process is used to realize the reserves in a distributed manner. This control process is embedded into the ARO problem, which is formulated as an affine ARO problem and then transformed into a deterministic optimization problem that is solved by off-shore solvers Apart from the reduction in the operation cost, the frequency restoration can be improved jointly, resulting in the coupled techno-economic contribution of the MGs in the coalition. The contribution of each MG is quantified using shapely value, a cooperative game approach. Simulations are conducted for a case study with 4 MGs and the results demonstrate the merits of the proposed cooperative scheduling scheme.

On the Optimality of Affine Policies for Budgeted Uncertainty Sets

In this paper, we study the performance of affine policies for a two-stage, adjustable, robust optimization problem with a fixed recourse and an uncertain right-hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets, widely used in practice, in which we can specify a budget on the adversarial deviations of the uncertain parameters from the nominal values to adjust the level of conservatism. The two-stage adjustable robust optimization problem is hard to approximate within a factor better than [Formula: see text] even for budget of uncertainty sets in which [Formula: see text] is the number of decision variables. Affine policies, in which the second-stage decisions are constrained to be an affine function of the uncertain parameters provide a tractable approximation for the problem and have been observed to exhibit good empirical performance. We show that affine policies give an [Formula: see text]-approximation for the two-stage, adjustable, robust problem with fixed nonnegative recourse for budgeted uncertainty sets. This matches the hardness of approximation, and therefore, surprisingly, affine policies provide an optimal approximation for the problem (up to a constant factor). We also show strong theoretical performance bounds for affine policy for a significantly more general class of intersection of budgeted sets, including disjoint constrained budgeted sets, permutation invariant sets, and general intersection of budgeted sets. Our analysis relies on showing the existence of a near-optimal, feasible affine policy that satisfies certain nice structural properties. Based on these structural properties, we also present an alternate algorithm to compute a near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.

Interplay of non-convex quadratically constrained problems with adjustable robust optimization

In this paper we explore convex reformulation strategies for non-convex quadratically constrained optimization problems (QCQPs). First we investigate such reformulations using Pataki’s rank theorem iteratively. We show that the result can be used in conjunction with conic optimization duality in order to obtain a geometric condition for the S-procedure to be exact. Based upon known results on the S-procedure, this approach allows for some insight into the geometry of the joint numerical range of the quadratic forms. Then we investigate a reformulation strategy introduced in recent literature for bilinear optimization problems which is based on adjustable robust optimization theory. We show that, via a similar strategy, one can leverage exact reformulation results of QCQPs in order to derive lower bounds for more complicated quadratic optimization problems. Finally, we investigate the use of reformulation strategies in order to derive characterizations of set-copositive matrix cones. Empirical evidence based upon first numerical experiments shows encouraging results.