03:30 PM - 03:55 PM
Inverse optimization of convex risk measures
Convex risk measures have emerged as a new standard for representing risk perception in optimization under uncertainty. Very little guidance however has been provided to date regarding the choice of a convex risk measure. In this work, we present a novel inverse optimization framework that imputes a convex risk measure based on the information of an observable solution and an initial estimate of the risk measure. Unlike classical inverse optimization, no parametric assumption is made about the risk measure. We show that somewhat surprisingly the inverse problems can always be reduced to finite-dimensional convex programs regardless of the complexity of the set of feasible solutions.
03:55 PM - 04:20 PM
Two-stage stochastic program with distributional ambiguity
In this talk, we develop a risk-averse two-stage stochastic program (RTSP) which explicitly incorporates the distributional ambiguity covering both discrete and continuous distributions. Starting from a set of historical data samples, we construct an ambiguity set for the probability distribution through nonparametric statistical estimation of its density function. We then formulate RTSP from the perspective of distributional robustness by hedging against the worst-case distribution within the ambiguity set and considering the corresponding expected total cost. In particular, we derive an equivalent reformulation for RTSP which explicitly reflects its linkage with a full spectrum of coherent risk measures under various risk-averseness levels. Our reformulation result can also be extended to other interesting stochastic programming models with expectation, conditional value-at-risk, chance, and stochastic dominance constraints. Furthermore, we perform convergence analysis to show that the ambiguity-averseness of RTSP vanishes as the data sample size grows to infinity, in the sense that the optimal objective value and the set of optimal solutions of RTSP converge to those of TSP. Finally, we develop a solution algorithm for the reformulation of RTSP based on the sample average approximation method, and numerical experiments on newsvendor and lot-sizing problems explain and demonstrate the effectiveness of our proposed framework.
04:20 PM - 04:45 PM
Robust chance constrained optimization in electricity networks with ramping constraints
We study the ramping costs and generation constraints in power networks in which renewable energy is the main source of uncertainty. A robust chance constrained optimization model is developed to design dispatch policies that manage the risk and maintain system reliability. Such model can be written using SOCP constraints for typical renewable power probability distributions. This model is compared in terms of efficiency with the stochastic programming formulation of the chance constraints using sampling techniques.
04:45 PM - 05:10 PM
Parallel scenario decomposition of risk averse 0-1 stochastic programs and chance-constrained 0-1 programs
In this talk, we discuss two recent projects on solving risk-averse 0-1 programs and chance-constrained 0-1 programs with scenario decomposition methods implemented in a distributed framework. For risk-averse programs, we consider coherent risk measures, and use the dual representations. We develop three variants of the scenario decomposition algorithm for solving equivalent minimax reformulations based on different relaxations of the nonanticipaticity constraints. The algorithms proceed by solving scenario subproblems to obtain candidate solutions and bounds, and subsequently cutting off the candidate solutions from the search space to achieve convergence to an optimal solution. For chance-constrained models, we derive a proposition that simplifies the computation of the Lagrangian dual in scenario decomposition, which allows replacing subgradient iterations with simple arithmetic, and significantly reduces the number of subproblems. We also implement cut aggregation and combinatorial optimization approaches. For both programs, we design three parallelization schemes trading off differently between communication time and computation time. Our results with variants of two stochastic 0-1 programming test instances demonstrate the scalability of the proposed decomposition and parallelization framework.