Convex approximations for a class of mixed-integer recourse models

We consider mixed-integer recourse (MIR) models with a single recourse constraint.We relate the secondstage value function of such problems to the expected simple integer recourse (SIR) shortage function. This allows to construct convex approximations for MIR problems by the same approach used for SIR models.

On the heterogeneous vehicle routing problem under demand uncertainty

In this paper we study the heterogeneous vehicle routing problem under demand uncertainty, on which there has been little research to our knowledge. The focus of the paper is to provide a strong formulation that also easily allows tractable robust and chance-constrained counterparts. To this end, we propose a basic Miller-Tucker-Zemlin (MTZ) formulation with the main advantage that uncertainty is restricted to the right-hand side of the constraints. This leads to compact and tractable counterparts of demand uncertainty. On the other hand, since the MTZ formulation is well known to provide a rather weak linear programming relaxation, we propose to strengthen the initial formulation with valid inequalities and lifting techniques and, furthermore, to dynamically add cutting planes that successively reduce the polyhedral region using a branch-and-cut algorithm. We complete our study with extensive computational analysis with different performance measures on different classes of instances taken from the literature. In addition, using simulation, we conduct a scenario-based risk level analysis for both cases where either unmet demand is allowed or not

Commitment and Dispatch of Heat and Power Units via Affinely Adjustable Robust Optimization

The joint management of heat and power systems is believed to be key to the integration of renewables into energy systems with a large penetration of district heating. Determining the day-ahead unit commitment and production schedules for these systems is an optimization problem subject to uncertainty stemming from the unpredictability of demand and prices for heat and electricity. Furthermore, owing to the dynamic features of production and heat storage units as well as to the length and granularity of the optimization horizon (e.g., one whole day with hourly resolution), this problem is in essence a multi-stage one. We propose a formulation based on robust optimization where recourse decisions are approximated as linear or piecewise-linear functions of the uncertain parameters. This approach allows for a rigorous modeling of the uncertainty in multi-stage decision-making without compromising computational tractability. We perform an extensive numerical study based on data from the Copenhagen area in Denmark, which highlights important features of the proposed model. Firstly, we illustrate commitment and dispatch choices that increase conservativeness in the robust optimization approach. Secondly, we appraise the gain obtained by switching from linear to piecewise-linear decision rules within robust optimization. Furthermore, we give directions for selecting the parameters defining the uncertainty set (size, budget) and assess the resulting trade-off between average profit and conservativeness of the solution. Finally, we perform a thorough comparison with competing models based on deterministic optimization and stochastic programming.Comment: 31 page

On multiple simple recourse models

We consider multiple simple recourse (MSR) models, both continuous and integer versions, which generalize the corresponding simple recourse (SR) models by allowing for a refined penalty cost structure for individual shortages and surpluses. It will be shown that (convex approximations of) such MSR models can be represented as explicitly specified continuous SR models, and thus can be solved efficiently by existing algorithms.

Stochastic Vehicle Routing with Recourse

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage optimization problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed -- but costs here become more expensive by a factor lambda. We present an O(log^2 n log(n lambda))-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of Theorem 1.

Approximation in stochastic integer programming

Approximation algorithms are the prevalent solution methods in the field of stochastic programming. Problems in this field are very hard to solve. Indeed, most of the research in this field has concentrated on designing solution methods that approximate the optimal solutions. However, efficiency in the complexity theoretical sense is usually not taken into account. Quality statements mostly remain restricted to convergence to an optimal solution without accompanying implications on the running time of the algorithms for attaining more and more accurate solutions. However, over the last twenty years also some studies on performance analysis of approximation algorithms for stochastic programming have appeared. In this direction we find both probabilistic analysis and worst-case analysis. There have been studies on performance ratios and on absolute divergence from optimality. Only recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most combinatorial optimization problems.