A Convex Approximation for Two-Stage Mixed-Integer Recourse Models with a Uniform Error Bound

We develop a convex approximation for two-stage mixed-integer recourse models, and we derive an error bound for this approximation that depends on the total variations of the probability density functions of the random variables in the model. We show that the error bound converges to zero if all these total variations converge to zero. Our convex approximation is a generalization of the one in Romeijnders, van der Vlerk, and Klein Haneveld [Math. Program., to appear] restricted to totally unimodular integer recourse models. For this special case it has the best worst-case error bound possible. The error bound in this paper is the first in the general setting of mixed-integer recourse models. As main building blocks in its derivation we generalize the asymptotic periodicity results of Gomory [Linear Algebra Appl., 2 (1969), pp. 451--558] for pure integer programs to the mixed-integer case, and we use the total variation error bounds on the expectation of periodic functions derived in Romeijnders, van der Vlerk, and Klein Haneveld [Math. Program., to appear].<br/

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.

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.

Parametric error bounds for convex approximations of two-stage mixed-integer recourse models with a random second-stage cost vector

We consider two-stage recourse models with integer restrictions in the second stage. These models are typically nonconvex and hence, hard to solve. There exist convex approximations of these models with accompanying error bounds. However, it is unclear how these error bounds depend on the distributions of the second-stage cost vector q.In this paper, we derive parametric error bounds whose dependence on the distribution of q is explicit: they scale linearly in the expected value of the `1-norm of q

Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

In traditional two-stage mixed-integer recourse models, the expected value of the total costs is minimized. In order to address risk-averse attitudes of decision makers, we consider a weighted mean-risk objective instead. Conditional value-at-risk is used as our risk measure. Integrality conditions on decision variables make the model non-convex and hence, hard to solve. To tackle this problem, we derive convex approximation models and corresponding error bounds, that depend on the total variations of the density functions of the random right-hand side variables in the model. We show that the error bounds converge to zero if these total variations go to zero. In addition, for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds.</p

Approximate and exact convexification approaches for solving two-stage mixed-integer recourse models

Many practical decision-making problems are subject to uncertainty. A powerful class of mathematical models designed for these problems is the class of mixed-integer recourse models. Such models have a wide range of applications in, e.g., healthcare, energy, and finance. They permit integer decision variables to accurately model, e.g., on/off restrictions or natural indivisibilities. The additional modelling flexibility of integer decision variables, however, comes at the expense of models that are significantly harder to solve. The reason is that including integer decision variables introduces non-convexity in the model, which poses a significant challenge for state-of-the-art solvers.In this thesis, we contribute to better decision making under uncertainty by designing efficient solution methods for mixed-integer recourse models. Our approach is to address the non-convexity caused by integer decision variables by using convexification. That is, we construct convex approximating models that closely approximate the original model. In addition, we derive performance guarantees for the solution obtained by solving the approximating model. Finally, we extensively test the solution methods that we propose and we find that they consistently outperform traditional solution methods on a wide range of benchmark instances

Convex approximations for complete integer recourse models

We consider convex approximations of the expected value function of a two-stage integer recourse problem. The convex approximations are obtained by perturbing the distribution of the random right-hand side vector. It is shown that the approximation is optimal for the class of problems with totally unimodular recourse matrices. For problems not in this class, the result is a convex lower bound that is strictly better than the one obtained from the LP relaxation.

Pragmatic convex approaches for risk-averse and distributionally robust mixed-integer recourse models

In this thesis I consider two classes of stochastic optimization models: risk-averse mixed-integer recourse (MIR) models and distributionally robust MIR models. These classes of models can be used to support decision making in situations where uncertainty about the future plays an important role. For example, one might need to invest in a new production facility while future demand for the produced goods is uncertain. Typically, these two classes of MIR models are non-convex as a result of the integer restrictions in the model. This makes these models extremely hard to solve from a computational point of view. My aim is to overcome this issue and find efficient solution approaches.In this thesis, I propose pragmatic convex approaches for risk-averse and distributionally robust MIR models. These approaches are based on the pragmatic idea of solving a convex model in order to find a reasonably good solution to the original, non-convex, model. For risk-averse MIR models this is achieved by constructing a convex approximation model, which can be solved efficiently. I prove that the approximation model is close to the original model in settings where probability distribution of the uncertain parameters is highly dispersed and I explicitly show the effect of the selected risk measure on the quality of the approximation. For distributionally robust MIR models I propose to restrict the uncertainty set to a class of special "convexifying" distributions. The resulting model is convex and, in particular settings, also resolves overfitting issues

Convex hull approximation of TU integer recourse models:Counterexamples, sufficient conditions, and special cases

We consider a convex approximation for integer recourse models. In particular, we showthat the claim of Van der Vlerk (2004) that this approximation yields the convex hull of totallyunimodular (TU) integer recourse models is incorrect. We discuss counterexamples, indicate which step of its proof does not hold in general, and identify a class of random variables for which the claim in Van der Vlerk (2004) is not true. At the same time, we derive additional assumptions under which the claim does hold. In particular, if the random variables in the model are independently and uniformly distributed, then these assumptions are satisfied