The impossibility theorem of fairness is a foundational result in the algorithmic fairness literature. It states that outside of special cases, one cannot exactly and simultaneously satisfy all three common and intuitive definitions of fairness demographic parity, equalized odds, and predictive rate parity. This result has driven most works to focus on solutions for one or two of the metrics.

In this paper, we focus on computing an NE that optimizes a given objective function.

There are other approaches (e.g., [ Here, if all team members play strategies according to an NE minimizing the adversary's utility, the Eq.(1c) ensures that binary variable This space is represented by Eq.(1), which involves nonlinear terms in Eq.(1a) Section 3.4 shows that our techniques can significantly reduce the time The procedure of CRM is shown in Algorithm 2, which is illustrated in Appendix A. A collection N of subsets of players is a binary collection if: 1. { i | i N } N ; Eqs.(1b)-(1g), (3), and (4) is the space of NEs. Example 1 provides an example of N .

In this paper, we focus on computing an NE that optimizes a given objective function.

This paper proposes a control-oriented optimization platform for autonomous mobile robots (AMRs), focusing on extending battery life while ensuring task completion. The requirement of fast AMR task planning while maintaining minimum battery state of charge, thus maximizing the battery life, renders a bilinear optimization problem. McCormick envelop technique is proposed to linearize the bilinear term. A novel planning algorithm with relaxed constraints is also developed to handle parameter uncertainties robustly with high efficiency ensured. Simulation results are provided to demonstrate the utility of the proposed methods in reducing battery degradation while satisfying task completion requirements.

Discretization-based methods have been proposed for solving nonconvex optimization problems with bilinear terms such as the pooling problem. These methods convert the original nonconvex optimization problems into mixed-integer linear programs (MILPs). In this paper we study tightening methods for these MILP models for the pooling problem, and derive valid constraints using upper bounds on bilinear terms. Computational results demonstrate the effectiveness of our methods in terms of reducing solution time.

The impossibility theorem of fairness is a foundational result in the algorithmic fairness literature. It states that outside of special cases, one cannot exactly and simultaneously satisfy all three common and intuitive definitions of fairness - demographic parity, equalized odds, and predictive rate parity. This result has driven most works to focus on solutions for one or two of the metrics. Rather than follow suit, in this paper we present a framework that pushes the limits of the impossibility theorem in order to satisfy all three metrics to the best extent possible. We develop an integer-programming based approach that can yield a certifiably optimal post-processing method for simultaneously satisfying multiple fairness criteria under small violations. We show experiments demonstrating that our post-processor can improve fairness across the different definitions simultaneously with minimal model performance reduction. We also discuss applications of our framework for model selection and fairness explainability, thereby attempting to answer the question: who's the fairest of them all?

A core challenge in policy optimization in competitive Markov decision processes is the design of efficient optimization methods with desirable convergence and stability properties. To tackle this, we propose competitive policy optimization (CoPO), a novel policy gradient approach that exploits the game-theoretic nature of competitive games to derive policy updates. Motivated by the competitive gradient optimization method, we derive a bilinear approximation of the game objective. In contrast, off-the-shelf policy gradient methods utilize only linear approximations, and hence do not capture interactions among the players. We instantiate CoPO in two ways:(i) competitive policy gradient, and (ii) trust-region competitive policy optimization. We theoretically study these methods, and empirically investigate their behavior on a set of comprehensive, yet challenging, competitive games. We observe that they provide stable optimization, convergence to sophisticated strategies, and higher scores when played against baseline policy gradient methods.