Stefan Scholtes

Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity

We study mathematical programs with complementarity constraints. Several stationarity concepts, based on a piecewise smooth formulation, are presented and compared. The concepts are related to stationarity conditions for certain smooth programs as well as to stationarity concepts for a nonsmooth exact penalty function. Further, we present Fiacco-McCormick type second order optimality conditions and an extension of the stability results of Robinson and Kojima to mathematical programs with complementarity constraints.

Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints

We study the convergence behavior of a sequence of stationary points of a parametric NLP which regularizes a mathematical program with equilibrium constraints (MPEC) in the form of complementarity conditions. Accumulation points are feasible points of the MPEC; they are C-stationary if the MPEC linear independence constraint qualification holds; they are M-stationary if, in addition, an approaching subsequence satisfies second order necessary conditions, and they are B-stationary if, in addition, an upper level strict complementarity condition holds. These results complement recent results of Fukushima and Pang [Convergence of a smoothing continuation method for mathematical programs with equilibrium constraints, in Ill-posed Variational Problems and Regularization Techniques, Springer-Verlag, New York, 1999]. We further show that every local minimizer of the MPEC which satisfies the linear independence, upper level strict complementarity, and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric NLP.

Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints

Recently, nonlinear programming solvers have been used to solve a range of mathematical programs with equilibrium constraints (MPECs). In particular, sequential quadratic programming (SQP) methods have been very successful. This paper examines the local convergence properties of SQP methods applied to MPECs. SQP is shown to converge superlinearly under reasonable assumptions near a strongly stationary point. A number of examples are presented that show that some of the assumptions are difficult to relax.

Introduction to piecewise differentiable equations

-1. Sample problems for nonsmooth equations. -2. Piecewise affline functions. -3. Elements from nonsmooth analysis. -4. Piecewise differentiable functions. -5. Sample applications.

A two-sided relaxation scheme for Mathematical Programs with Equilibrium Constraints

We propose a relaxation scheme for mathematical programs with equilibrium constraints (MPECs). In contrast to previous approaches, our relaxation is two-sided: both the complementarity and the nonnegativity constraints are relaxed. The proposed relaxation update rule guarantees (under certain conditions) that the sequence of relaxed subproblems will maintain a strictly feasible interior---even in the limit. We show how the relaxation scheme can be used in combination with a standard interior-point method to achieve superlinear convergence. Numerical results on the MacMPEC test problem set demonstrate the fast local convergence properties of the approach.

How Stringent Is the Linear Independence Assumption for Mathematical Programs with Complementarity Constraints

The linear independence constraint qualifications (LICQ) plays an important role in the analysis of mathematical programs with complementarity constraints (MPCCs) and is a vital ingredient to convergence analyses of SQP-type or smoothing methods, cf., e.g., Fukushima and Pang (1999), Luo et al. (1996), Scholtes and StA¶hr (1999), Scholtes (2001), StA¶hr (2000). We will argue in this paper that LICQ is not a particularly stringent assumption for MPCCs. Our arguments are based on an extension of Jongens (1977) genericity analysis to MPCCs. His definitions of nondegenerate critical points and regular programs extend naturally to MPCCs and his genericity results generalize straightforwardly to MPCCs in standard form. An extension is not as straightforward for MPCCs with the particular structure induced by lower-level stationarity conditions for variational inequalities or optimization problems. We show that LICQ remains a generic property for this class of MPCCs.

Stress on the Ward: Evidence of Safety Tipping Points in Hospitals

Do hospitals experience safety tipping points as utilization increases, and if so, what are the implications for hospital operations management? We argue that safety tipping points occur when managerial escalation policies are exhausted and workload variability buffers are depleted. Front-line clinical staff is forced to ration resources and, at the same time, becomes more error prone as a result of elevated stress hormone levels. We confirm the existence of safety tipping points for in-hospital mortality using the discharge records of 82,280 patients across six high-mortality-risk conditions from 256 clinical departments of 83 German hospitals. Focusing on survival during the first seven days following admission, we estimate a mortality tipping point at an occupancy level of 92.5%. Among the 17% of patients in our sample who experienced occupancy above the tipping point during the first seven days of their hospital stay, high occupancy accounted for one in seven deaths. The existence of a safety tipping point has important implications for hospital management. First, flexible capacity expansion is more cost-effective for safety improvement than rigid capacity, because it will only be used when occupancy reaches the tipping point. In the context of our sample, flexible staffing saves more than 40% of the cost of a fully staffed capacity expansion, while achieving the same reduction in mortality. Second, reducing the variability of demand by pooling capacity in hospital clusters can greatly increase safety in a hospital system, because it reduces the likelihood that a patient will experience occupancy levels beyond the tipping point. Pooling the capacity of nearby hospitals in our sample reduces the number of deaths due to high occupancy by 34%. This paper was accepted by Serguei Netessine, operations management.

Minimal pairs of convex bodies in two dimensions

In [7] the notion of minimal pairs of convex compact subsets of a Hausdorff topological vector space was introduced and it was conjectured, that minimal pairs in an equivalence class of the Hormander-Radstrom lattice are unique up to translation. We prove this statement for the two-dimensional case. To achieve this we prove a necessary and sufficient condition in terms of mixed volumes that a translate of a convex body in ℝ n is contained in another convex body. This generalizes a theorem of Weil ( cf . [10]).

Sensitivity analysis of composite piecewise smooth equations

AbstractThis paper is a contribution to the sensitivity analysis of piecewise smooth equations. A piecewise smooth function is a Lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible B-derivative at this point. We emphasise the role of functions of the typef=g °h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf=g °