The concept of Nash equilibrium enlightens the structure of rational behavior in multi-agent settings. However, the concept is as helpful as one may compute it efficiently. We introduce the Cut-and-Play, an algorithm to compute Nash equilibria for non-cooperative simultaneous games where each player's objective is linear in their variables and bilinear in the other players' variables. Using the rich theory of integer programming, we alternate between constructing (i.) increasingly tighter outer approximations of the convex hull of each player's feasible set -- by using branching and cutting plane methods -- and (ii.) increasingly better inner approximations of these hulls -- by finding extreme points and rays of the convex hulls. In particular, we prove the correctness of our algorithm when these convex hulls are polyhedra. Our algorithm allows us to leverage the mixed integer programming technology to compute equilibria for a large class of games. Further, we integrate existing cutting plane families inside the algorithm, significantly speeding up equilibria computation. We showcase a set of extensive computational results for Integer Programming Games and simultaneous games among bilevel leaders. In both cases, our framework outperforms the state-of-the-art in computing time and solution quality.
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We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error $1-r/f$ on the spectral interval of $A$. By minimizing this upper bound over all interpolation points, we obtain a new, simple and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant $r$. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating $r(A)$, where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for $r$ following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of $r$ leading to a small error, even in presence of finite precision arithmetic.
In this paper, we study a non-local approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the non-local problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and non-local problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the non-local problem and that of the local one, both in continuous-time and Backward Euler time discretization. We then turn to studying continuum limits of non-local problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows, then almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.
A Regret Minimizing Set (RMS) is a useful concept in which a smaller subset of a database is selected while mostly preserving the best scores along every possible utility function. In this paper, we study the $k$-Regret Minimizing Sets ($k$-RMS) and Average Regret Minimizing Sets (ARMS) problems. $k$-RMS selects $r$ records from a database such that the maximum regret ratio between the $k$-th best score in the database and the best score in the selected records for any possible utility function is minimized. Meanwhile, ARMS minimizes the average of this ratio within a distribution of utility functions. Particularly, we study approximation algorithms for $k$-RMS and ARMS from the perspective of approximating the happiness ratio, which is equivalent to one minus the regret ratio. In this paper, we show that the problem of approximating the happiness of a $k$-RMS within any finite factor is NP-Hard when the dimensionality of the database is unconstrained and extend the result to an inapproximability proof for the regret. We then provide approximation algorithms for approximating the happiness of ARMS with better approximation ratios and time complexities than known algorithms for approximating the regret. We further provide dataset reduction schemes which can be used to reduce the runtime of existing heuristic based algorithms, as well as to derive polynomial-time approximation schemes for $k$-RMS when dimensionality is fixed. Finally, we provide experimental validation.
The standard game-theoretic solution concept, Nash equilibrium, assumes that all players behave rationally. If we follow a Nash equilibrium and opponents are irrational (or follow strategies from a different Nash equilibrium), then we may obtain an extremely low payoff. On the other hand, a maximin strategy assumes that all opposing agents are playing to minimize our payoff (even if it is not in their best interest), and ensures the maximal possible worst-case payoff, but results in exceedingly conservative play. We propose a new solution concept called safe equilibrium that models opponents as behaving rationally with a specified probability and behaving potentially arbitrarily with the remaining probability. We prove that a safe equilibrium exists in all strategic-form games (for all possible values of the rationality parameters), and prove that its computation is PPAD-hard. We present exact algorithms for computing a safe equilibrium in both 2 and $n$-player games, as well as scalable approximation algorithms.
Financial networks model a set of financial institutions (firms) interconnected by obligations. Recent work has introduced to this model a class of obligations called credit default swaps, a certain kind of financial derivatives. The main computational challenge for such systems is known as the clearing problem, which is to determine which firms are in default and to compute their exposure to systemic risk, technically known as their recovery rates. It is known that the recovery rates form the set of fixed points of a simple function, and that these fixed points can be irrational. Furthermore, Schuldenzucker et al. (2016) have shown that finding a weakly (or "almost") approximate (rational) fixed point is PPAD-complete. We further study the clearing problem from the point of view of irrationality and approximation strength. Firstly, we observe that weakly approximate solutions may misrepresent the actual financial state of an institution. On this basis, we study the complexity of finding a strongly (or "near") approximate solution, and show FIXP-completeness. We then study the structural properties required for irrationality, and we give necessary conditions for irrational solutions to emerge: The presence of certain types of cycles in a financial network forces the recovery rates to take the form of roots of non-linear polynomials. In the absence of a large subclass of such cycles, we study the complexity of finding an exact fixed point, which we show to be a problem close to, albeit outside of, PPAD.
This paper deals with robust inference for parametric copula models. Estimation using Canonical Maximum Likelihood might be unstable, especially in the presence of outliers. We propose to use a procedure based on the Maximum Mean Discrepancy (MMD) principle. We derive non-asymptotic oracle inequalities, consistency and asymptotic normality of this new estimator. In particular, the oracle inequality holds without any assumption on the copula family, and can be applied in the presence of outliers or under misspecification. Moreover, in our MMD framework, the statistical inference of copula models for which there exists no density with respect to the Lebesgue measure on $[0,1]^d$, as the Marshall-Olkin copula, becomes feasible. A simulation study shows the robustness of our new procedures, especially compared to pseudo-maximum likelihood estimation. An R package implementing the MMD estimator for copula models is available.
We present a two-level branch-and-bound (BB) algorithm to compute the optimal gripper pose that maximizes a grasp metric in a restricted search space. Our method can take the gripper's kinematics feasibility into consideration to ensure that a given gripper can reach the set of grasp points without collisions or predict infeasibility with finite-time termination when no pose exists for a given set of grasp points. Our main technical contribution is a novel mixed-integer conic programming (MICP) formulation for the inverse kinematics of the gripper that uses a small number of binary variables and tightened constraints, which can be efficiently solved via a low-level BB algorithm. Our experiments show that optimal gripper poses for various target objects can be computed taking 20-180 minutes of computation on a desktop machine and the computed grasp quality, in terms of the Q1 metric, is better than those generated using sampling-based planners.
Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.
In warehouses, order picking is known to be the most labor-intensive and costly task in which the employees account for a large part of the warehouse performance. Hence, many approaches exist, that optimize the order picking process based on diverse economic criteria. However, most of these approaches focus on a single economic objective at once and disregard ergonomic criteria in their optimization. Further, the influence of the placement of the items to be picked is underestimated and accordingly, too little attention is paid to the interdependence of these two problems. In this work, we aim at optimizing the storage assignment and the order picking problem within mezzanine warehouse with regards to their reciprocal influence. We propose a customized version of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) for optimizing the storage assignment problem as well as an Ant Colony Optimization (ACO) algorithm for optimizing the order picking problem. Both algorithms incorporate multiple economic and ergonomic constraints simultaneously. Furthermore, the algorithms incorporate knowledge about the interdependence between both problems, aiming to improve the overall warehouse performance. Our evaluation results show that our proposed algorithms return better storage assignments and order pick routes compared to commonly used techniques for the following quality indicators for comparing Pareto fronts: Coverage, Generational Distance, Euclidian Distance, Pareto Front Size, and Inverted Generational Distance. Additionally, the evaluation regarding the interaction of both algorithms shows a better performance when combining both proposed algorithms.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
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