Global Robust Stability in a General Price and Assortment Competition Model

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the “regular” extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models, such as the multinomial logit model or any of its variants. We show that a special equilibrium in this model has global robust stability. This means that, from any starting point, the market converges to this equilibrium when firms use a particular response mapping to dynamically adjust their own prices in response to their competitors’ prices. The mapping requires each firm to only know the demand function and cost structure for its own products (but not for other firms’ products).

Positivity Preserving Gradient Approximation with Linear Finite Elements

Preserving positivity precludes that linear operators onto continuous piecewise affine functions provide near best approximations of gradients. Linear interpolation thus does not capture the approximation properties of positive continuous piecewise affine functions. To remedy, we assign nodal values in a nonlinear fashion such that their global best error is equivalent to a suitable sum of local best errors with positive affine functions. As one of the applications of this equivalence, we consider the linear finite element solution to the elliptic obstacle problem and derive that its error is bounded in terms of these local best errors.

Approximation of fracture energies with p-growth via piecewise affine finite elements

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBDp(Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBDp functions, for p ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C1 hypersurfaces. The strains of the approximating functions converge strongly in Lp to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudenthal partition of a cubic grid.

Solution uniqueness of convex piecewise affine functions based optimization with applications to constrained ℓ1 minimization

In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems incorporates many important polyhedral constrained ℓ1 recovery problems arising from sparse optimization, such as basis pursuit, LASSO, and basis pursuit denoising, as well as polyhedral gauge recovery. By leveraging the max-formulation of convex piecewise affine functions and convex analysis tools, we develop dual variables based necessary and sufficient uniqueness conditions via simple and yet unifying approaches; these conditions are applied to a wide range of ℓ1 minimization problems under possible polyhedral constraints. An effective linear program based scheme is proposed to verify solution uniqueness conditions. The results obtained in this paper not only recover the known solution uniqueness conditions in the literature by removing restrictive assumptions but also yield new uniqueness conditions for much broader constrained ℓ1-minimization problems.

Parametric Strategy Iteration

Program behavior may depend on parameters, which are either configuredbefore compilation time, or provided at runtime, e.g., by sensors or other input devices.Parametric program analysis explores how different parameter settings may affect theprogram behavior.In order to infer invariants depending on parameters, we introduce parametric strategy iteration.This algorithm determines the precise least solution of systems of integer equations dependingon surplus parameters. Conceptually, our algorithm performs ordinary strategy iterationon the given integer system for all possible parameter settings in parallel.This is made possible by means of region trees to represent the occurring piecewise affine functions.We indicate that each required operation on these trees is polynomial-time if only constantly manyparameters are involved.Parametric strategy iteration for systems of integer equationsallows to construct parametric integer interval analysisas well as parametric analysis of differences of integer variables.It thus provides a general technique to realize precise parametricprogram analysis if numerical properties of integer variables are of concern.