On Topological Properties of Min-Max Functions

We examine the topological structure of the upper-level set M ^max given by a min-max function φ. It is motivated by recent progress in Generalized Semi-Infinite Programming (GSIP). Generically, M ^max is proven to be the topological closure of the GSIP feasible set (see Guerra-Vázquez et al. 2009; Günzel et al., Cent Eur J Oper Res 15(3):271–280, 2007). We formulate two assumptions (Compactness Condition CC and Sym-MFCQ) which imply that M ^max is a Lipschitz manifold (with boundary). The Compactness Condition is shown to be stable under C ⁰-perturbations of the defining functions of φ. Sym-MFCQ can be seen as a constraint qualification in terms of Clarke’s subdifferential of the min-max function φ. Moreover, Sym-MFCQ is proven to be generic and stable under C ¹-perturbations of the defining functions which fulfill the Compactness Condition. Finally we apply our results to GSIP and conclude that generically the closure of the GSIP feasible set is a Lipschitz manifold (with boundary).     

Network manipulation algorithm based on inexact alternating minimization

In this paper, we present a network manipulation algorithm based on an alternating minimization scheme from Nesterov (Soft Comput 1–12, 2020). In our context, the alternative process mimics the natural behavior of agents and organizations operating on a network. By selecting starting distributions, the organizations determine the short-term dynamics of the network. While choosing an organization in accordance with their manipulation goals, agents are prone to errors. This rational inattentive behavior leads to discrete choice probabilities. We extend the analysis of our algorithm to the inexact case, where the corresponding subproblems can only be solved with numerical inaccuracies. The parameters reflecting the imperfect behavior of agents and the credibility of organizations, as well as the condition number of the network transition matrix have a significant impact on the convergence of our algorithm. Namely, they turn out not only to improve the rate of convergence, but also to reduce the accumulated errors. From the mathematical perspective, this is due to the induced strong convexity of an appropriate potential function.

     

Hexosaminidase inhibitors as new drug candidates for the therapy of osteoarthritis.

BACKGROUND: Articular cartilage from patients with osteoarthritis is characterized by a decreased concentration and reduced size of glycosaminoglycans. Degeneration of the cartilage matrix is a multifactorial process, which is due in part to accelerated glycosaminoglycan catabolism. Recently, we have demonstrated that hexosaminidase represents the dominant glycosaminoglycan-degrading glycosidase released by chondrocytes into the extracellular compartment and is the dominant glycosidase in synovial fluid from patients with osteoarthritis. Inhibition of hexosaminidase activity may represent a novel approach to the prevention of cartilage matrix glycosaminoglycan degradation and a potentially new strategy to treat osteoarthritis. RESULTS: We have synthesized and investigated a series of iminocyclitols designed as transition-state analog inhibitors of human hexosaminidase, and demonstrated that the five-membered iminocyclitol 4 expresses the strongest inhibitory activity with K(i)=24 nM. Inhibition of hexosaminidase activity in human cultured articular chondrocytes and human chondrosarcoma cells with iminocyclitol 4 resulted in accumulation of hyaluronic acid and sulfated glycosaminoglycans in the cell-associated fraction. Similarly, incubation of human cartilage tissue with iminocyclitol 4 resulted in an accumulation of glycosaminoglycans in the pericellular compartment. CONCLUSIONS: Inhibition of hexosaminidase activity represents a new strategy for preventing or even reversing cartilage degradation in patients with osteoarthritis.     

Topological Approach to Mathematical Programs with Switching Constraints

Set-Valued and Variational Analysis - We study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved....     

Constrained Optimization: Projected Gradient Flows

We consider a dynamical system approach to solve finite-dimensional smooth optimization problems with a compact and connected feasible set. In fact, by the well-known technique of equalizing inequality constraints using quadratic slack variables, we transform a general optimization problem into an associated problem without inequality constraints in a higher-dimensional space. We compute the projected gradient for the latter problem and consider its projection on the feasible set in the original, lower-dimensional space. In this way, we obtain an ordinary differential equation in the original variables, which is specially adapted to treat inequality constraints (for the idea, see Jongen and Stein, Frontiers in Global Optimization, pp. 223–236, Kluwer Academic, Dordrecht, 2003). The article shows that the derived ordinary differential equation possesses the basic properties which make it appropriate to solve the underlying optimization problem: the longtime behavior of its trajectories becomes stationary, all singularities are critical points, and the stable singularities are exactly the local minima. Finally, we sketch two numerical methods based on our approach.     

Characterization of strong stability for C-stationary points in MPCC

We study mathematical programs with complementarity constraints (MPCC). Special focus will be on C-stationary points. Under the Linear Independence Constraint Qualification we characterize strong stability of C-stationary points (in the sense of Kojima) by means of first and second order information of the defining functions. It turns out that strong stability of C-stationary points allows a possible degeneracy of bi-active Lagrange multipliers. Some relations to other stationarity concepts (such as A-, M-, S- and B-stationarity) are shortly discussed.     

On the local representation of piecewise smooth equations as a Lipschitz manifold

We study systems of equations, F(x)=0$F(x)=0$, given by piecewise differentiable functions F:Rⁿ→R^k$F:{\mathbb{R}}^n\to {\mathbb{R}}^k$, k?n$k?n$. The focus is on the representability of the solution set locally as an (n−k)$(n-k)$-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarke?s generalized Jacobians. The SFRA claims the existence of a basis in which Clarke?s inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarke?s generalized Jacobian. The article is devoted to the conjectured equivalence of SFRA and FRA. For min-type functions, we give reformulations of SFRA and FRA using orthogonal projections, basis enlargements, cross products, dual variables, as well as via exponentially many convex cones. The equivalence of SFRA and FRA is shown to be true for min-type functions in the new case k=3$k=3$.     

Bilevel optimization: on the structure of the feasible set

We consider bilevel optimization from the optimistic point of view. Let the pair (x, y) denote the variables. The main difficulty in studying such problems lies in the fact that the lower level contains a global constraint. In fact, a point (x, y) is feasible if y solves a parametric optimization problem L(x). In this paper we restrict ourselves to the special case that the variable x is one-dimensional. We describe the generic structure of the feasible set M. Moreover, we discuss local reductions of the bilevel problem as well as corresponding optimality criteria. Finally, we point out typical problems that appear when trying to extend the ideas to higher dimensional x-dimensions. This will clarify the high intrinsic complexity of the general generic structure of the feasible set M and corresponding optimality conditions for the bilevel problem U.