Multi-objective optimization aims to find a set of solutions that achieve the best trade-off among multiple conflicting objective functions. While various multi-objective optimization algorithms have been proposed so far, most of them aim to find finite solutions as an approximation of the Pareto set, which may not adequately capture the entire structure of the Pareto set, especially when the number of variables is large. To overcome this limitation, we propose a method to obtain a parametric hypersurface representing the entire Pareto set instead of a finite set of points. Since the Pareto set of an $M$-objective optimization problem typically forms an $(M-1)$-dimensional simplex, we use a B{é}zier simplex as a model to express the Pareto set. We then develop a stochastic gradient descent-based algorithm that updates the B{é}zier simplex model toward the Pareto set, introducing a preconditioning matrix to enhance convergence. Our convergence analysis demonstrated that the proposed algorithm outperforms naive stochastic gradient descent in terms of convergence rate. Furthermore, we validate the effectiveness of our method through various multi-objective optimization problem instances, including real-world problems.

Contextual Inverse Optimization (CIO) is a generalized framework of the predict-then-optimize approach, also referred to as decision-focused learning or contextual optimization, aiming to learn a model that predicts the unknown parameters of a nominal optimization problem using related covariates without compromising the solution quality. Unlike the predict-then-optimize approach, which assumes access to datasets containing realized unknown parameters, CIO considers a setting where only historical optimal solutions are available. Previous work has primarily focused on CIO under linear programming problems and proposed methods based on optimality conditions. In this study, we propose a general algorithm based on inverse optimization as a more general approach that does not require optimality conditions. To validate its effectiveness, we apply the proposed method to multiple CIO problems and demonstrate that it performs comparably to or better than existing predict-then-optimize methods, even without ground-truth unknown parameters.

We consider the quantum kernelized bandit problem, where the player observes information of rewards through quantum circuits termed the quantum reward oracle, and the mean reward function belongs to a reproducing kernel Hilbert space (RKHS). We propose a UCB-type algorithm that utilizes the quantum Monte Carlo (QMC) method and provide regret bounds in terms of the decay rate of eigenvalues of the Mercer operator of the kernel. Our algorithm achieves $\widetilde{O}\left( T^{\frac{3}{1 + \beta_p}} \log\left(\frac{1}{\delta} \right)\right)$ and $\widetilde{O} \left( \log^{3(1 + \beta_e^{-1})/2} (T) \log\left(\frac{1 }{\delta} \right) \right)$ cumulative regret bounds with probability at least $1-\delta$ if the kernel has a $\beta_p$-polynomial eigendecay and $\beta_e$-exponential eigendecay, respectively. In particular, in the case of the exponential eigendecay, our regret bounds exponentially improve that of classical algorithms. Moreover, our results indicate that our regret bound is better than the lower bound in the classical kernelized bandit problem if the rate of decay is sufficiently fast.