In this work, we combine ideas from the fields of Semi-Definite Programming (SDP) and Multi-Objective Programming (MOP) to compute the Pareto
frontier of combinatorial optimization (CO) problems. Our focus is on the Single-Row Facility Layout Problem (SRFLP). The SRFLP is a
well-known NP-hard CO problem in the field of industrial engineering. The problem consists of finding the optimal arrangement of
facilities along a single row, such that the total cost is minimized. The SRFLP is frequently extended to a multi-objective optimization
problem, to simultaneously consider conflicting objectives, such as flow distance, CO2 emissions or total material handling cost. We
propose a novel approach that leverages the power of SDP to solve the SRFLP with multiple objectives. Specifically, our focus is to
combine objective space techniques, such as the weighted sum method and the epsilon-constraint method, with SDP. Solution algorithms
based on these techniques are used in a computational study to compare our novel approach with objective space methods using integer
programming formulations. For this comparison, we consider an adapted set of benchmark instances from the literature.

Network games provide a powerful framework for modeling agent interactions in networked systems, where players are represented by nodes in a graph, and their payoffs depend on the actions taken by their neighbors. Extending the framework of network games, in this work we introduce and study semidefinite network games. In this model, each player selects a positive semidefinite matrix with trace equal to one, known as a density matrix,
to engage in a two-player game with every neighboring node.
The player's payoff is the cumulative payoff acquired from these edge games. Network semidefinite games are of interest because they provide a simplified framework for representing quantum strategic interactions.
Initially, we focus on the zero-sum setting, where the sum of all players' payoffs is equal to zero. We establish that in this class of games, Nash equilibria can be characterized as the projection of a spectrahedron.
Furthermore, we demonstrate that determining whether a game is a semidefinite network game is equivalent to deciding if the value of a semidefinite program is zero. Beyond the zero-sum case, we characterize Nash equilibria as the solutions of a semidefinite linear complementarity problem.

Quantum potential games and quantum common-interest games model quantum agents acting cooperatively to maximize a common utility function. In this presentation we shall introduce the setting of quantum common-interest games where players have density matrices as strategies and seek to maximize a common multi-linear utility function, which is a setting that generalizes classical common-interest games. We shall then study various decoupled learning dynamics for learning in such games and explore their properties. Decoupled learning dynamics for quantum common-interest games can be used as algorithms for linear optimization over the set of separable states, which is otherwise known as the Best Separable State problem.

Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero- sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $O(d/\epsilon^2)$ iterations to $\epsilon$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $O(d/\epsilon)$ iterations to $\epsilon$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous’ original algorithm sets a new benchmark for computing $\epsilon$-Nash equilibria in quantum zero-sum games.