Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic

In this paper, we present a multistage time consistent Expected Conditional Risk Measure for minimizing a linear combination of the expected mean and the expected variance, so-called Expected Mean-Variance. The model is formulated as a multistage stochastic mixed-integer quadratic programming problem combining risk-sensitive cost and scenario analysis approaches. The proposed problem is solved by a matheuristic based on the Branch-and-Fix Coordination method. The multistage scenario cluster primal decomposition framework is extended to deal with large-scale quadratic optimization by means of stage-wise reformulation techniques. A specific case study in risk-sensitive production planning is used to illustrate that a remarkable decrease in the expected variance (risk cost) is obtained. A competitive behavior on the part of our methodology in terms of solution quality and computation time is shown when comparing with plain use of CPLEX in 150 benchmark instances, ranging up to 711,845 constraints and 193,000 binary variables.project MTM2015-65317-P (MINECO/FEDER/EU); BERC 2014-2017; IT-928-16; and by the University of the Basque Country UPV/EHU; BCAM Severo Ochoa excellence accreditation Grant SEV-2013-0323; BERC 2014-201

Distributionally robust optimization through the lens of submodularity

Distributionally robust optimization is used to solve decision making problems under adversarial uncertainty where the distribution of the uncertainty is itself ambiguous. In this paper, we identify a class of these instances that is solvable in polynomial time by viewing it through the lens of submodularity. We show that the sharpest upper bound on the expectation of the maximum of affine functions of a random vector is computable in polynomial time if each random variable is discrete with finite support and upper bounds (respectively lower bounds) on the expected values of a finite set of submodular (respectively supermodular) functions of the random vector are specified. This adds to known polynomial time solvable instances of the multimarginal optimal transport problem and the generalized moment problem by bridging ideas from convexity in continuous optimization to submodularity in discrete optimization. In turn, we show that a class of distributionally robust optimization problems with discrete random variables is solvable in polynomial time using the ellipsoid method. When the submodular (respectively supermodular) functions are structured, the sharp bound is computable by solving a compact linear program. We illustrate this in two cases. The first is a multimarginal optimal transport problem with given univariate marginal distributions and bivariate marginals satisfying specific positive dependence orders along with an extension to incorporate higher order marginal information. The second is a discrete moment problem where a set of marginal moments of the random variables are given along with lower bounds on the cross moments of pairs of random variables. Numerical experiments show that the bounds improve by 2 to 8 percent over bounds that use only univariate information in the first case, and by 8 to 15 percent over bounds that use the first moment in the second case.Comment: 36 Pages, 6 Figure

Applying stochastic programming models in financial risk management

This research studies two modelling techniques that help seek optimal strategies in financial risk management. Both are based on the stochastic programming methodology. The first technique is concerned with market risk management in portfolio selection problems; the second technique contributes to operational risk management by optimally allocating workforce from a managerial perspective. The first model involves multiperiod decisions (portfolio rebalancing) for an asset and liability management problem and deals with the usual uncertainty of investment returns and future liabilities. Therefore it is well-suited to a stochastic programming approach. A stochastic dominance concept is applied to control the risk of underfunding. A small numerical example and a backtest are provided to demonstrate advantages of this new model which includes stochastic dominance constraints over the basic model. Adding stochastic dominance constraints comes with a price: it complicates the structure of the underlying stochastic program. Indeed, new constraints create a link between variables associated with different scenarios of the same time stage. This destroys the usual tree-structure of the constraint matrix in the stochastic program and prevents the application of standard stochastic programming approaches such as (nested) Benders decomposition and progressive hedging. A structure-exploiting interior point method is applied to this problem. Computational results on medium scale problems with sizes reaching about one million variables demonstrate the efficiency of the specialised solution technique. The second model deals with operational risk from human origin. Unlike market risk that can be handled in a financial manner (e.g. insurances, savings, derivatives), the treatment of operational risks calls for a “managerial approach”. Consequently, we propose a new way of dealing with operational risk, which relies on the well known Aggregate Planning Model. To illustrate this idea, we have adapted this model to the case of a back office of a bank specialising in the trading of derivative products. Our contribution corresponds to several improvements applied to stochastic programming modelling. First, the basic model is transformed into a multistage stochastic program in order to take into account the randomness associated with the volume of transaction demand and with the capacity of work provided by qualified and non-qualified employees over the planning horizon. Second, as advocated by Basel II, we calculate the probability distribution based on a Bayesian Network to circumvent the difficulty of obtaining data which characterises uncertainty in operations. Third, we go a step further by relaxing the traditional assumption in stochastic programming that imposes a strict independence between the decision variables and the random elements. Comparative results show that in general these improved stochastic programming models tend to allocate more human expertise in order to hedge operational risks. The dual solutions of the stochastic programs are exploited to detect periods and nodes that are at risk in terms of expertise availability