Traditional distance metric learning with side information usually formulates the objectives using the covariance matrices of the data point pairs in the two constraint sets of must-links and cannotlinks. Because the covariance matrix computes the sum of the squared l2-norm distances, it is prone to both outlier samples and outlier features. To develop a robust distance metric learning method, ...

In game theory and in particular in the theory of stochastic games, we encounter systems of polynomial equalities and inequalities. We start with a few examples. The first example relates to the minmax value and optimal strategies of a two-person zero-sum game with finitely many strategies. Consider a twoperson zero-sum game represented by a k ×m matrix A = (aij), 1 ≤ i ≤ k and 1 ≤ j ≤ m. The n...

This paper investigates, for the first time in the literature, the approximation of minmax (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a constant number of scenarios, we establish fully polynomial-time approximation schemes for the min-max versions of these problems, using relationships between multi-objective and min-max optimization. Us...

The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NP-hard and Sqrt-Sum-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in strategic form with integer payoffs is trembling hand perfect. Analogous results are shown for a number of other solution concepts, including proper equilibrium, (the strategy part of) sequentia...

Today, IEEE 802.11 wireless LANs (WLANs) are widely deployed throughout the world. It is known that static channel assignments are not sufficient for these networks due to traffic variations. However, in order to maintain compatibility with the existing protocol standards, it is not possible to employ dynamic channel allocation methods to handle traffic-load fluctuations. Rather, adaptive assig...

In this paper, we provide a generic anytime lower bounding procedure for minmax regret optimization problems. We show that the lower bound obtained is always at least as accurate as the lower bound recently proposed by Chassein and Goerigk [2]. The validity of the bound is based on game theoretic arguments and its computation is performed via a double oracle algorithm [6] that we specify. The l...

In graph clustering methods, MinMax Cut tends to provide more balanced clusters as compared to Ratio Cut and Normalized Cut. The traditional approach used spectral relaxation to solve the graph cut problem. The main disadvantage of this approach is that the obtained spectral solution has mixed signs, which could severely deviate from the true solution and have to resort to other clustering meth...

We consider the problems of finding two optimal triangulations of a convex polygon: MaxMin area and MinMax area. These are the triangulations that maximize the area of the smallest area triangle in a triangulation, and respectively minimize the area of the largest area triangle in a triangulation, over all possible triangulations. The problem was originally solved by Klincsek by dynamic program...

The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms o...

Normally distributed data arises in various contexts and often one is interested in estimating its variance. The authors limit themselves in this chapter to the class of estimators that are (positive) multiples of sample variances. Two important qualities of estimators are bias and variance, which respectively capture the estimator’s accuracy and precision. Apart from the two classical estimato...