Abstract: Using recently developed adjoint methods for computing the shape derivatives of functions that depend on MHD equilibria (Antonsen et al. 2019; Paul et al. 2020), we present the first example of analytic gradient-based optimization of fixed-boundary stellarator equilibria. We take advantage of gradient information to optimize figures of merit of relevance for stellarator design, including the rotational transform, magnetic well, and quasisymmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space ( 50 500) in comparison with a finite-difference gradient-based method. We discuss regularization objectives of relevance for fixed-boundary optimization, including a novel method that prevents self-intersection of the plasma boundary. Abstract: In atomic, molecular, and nuclear physics, the method of complex coordinate rotation is a widely used theoretical tool for studying resonant states.

This paper presents a triple optimization algorithm of two-dimensional space, driving path and driving speed, and iterates in the time dimension to obtain the local optimal solution of path and speed in the optimal driving area. Design iterative algorithm to solve the best driving path and speed within the limited conditions. The algorithm can meet the path planning needs of automatic driving vehicle in complex scenes and medium and high-speed scenes.

Stochastic compositional optimization (SCO) has attracted considerable attention because of its broad applicability to important real-world problems. However, existing works on SCO assume that the projection within a solution update is simple, which fails to hold for problem instances where the constraints are in the form of expectations, such as empirical conditional value-at-risk constraints. We study a novel model that incorporates single-level expected value and two-level compositional constraints into the current SCO framework. Our model can be applied widely to data-driven optimization and risk management, including risk-averse optimization and high-moment portfolio selection, and can handle multiple constraints. We further propose a class of primal-dual algorithms that generates sequences converging to the optimal solution at the rate of $\cO(\frac{1}{\sqrt{N}})$under both single-level expected value and two-level compositional constraints, where $N$ is the iteration counter, establishing the benchmarks in expected value constrained SCO.

Fairness-aware recommender systems that have a provider-side fairness concern seek to ensure that protected group(s) of providers have a fair opportunity to promote their items or products. There is a ``cost of fairness'' borne by the consumer side of the interaction when such a solution is implemented. This consumer-side cost raises its own questions of fairness, particularly when personalization is used to control the impact of the fairness constraint. In adopting a personalized approach to the fairness objective, researchers may be opening their systems up to strategic behavior on the part of users. This type of incentive has been studied in the computational social choice literature under the terminology of ``bossiness''. The concern is that a bossy user may be able to shift the cost of fairness to others, improving their own outcomes and worsening those for others. This position paper introduces the concept of bossiness, shows its application in fairness-aware recommendation and discusses strategies for reducing this strategic incentive.

Predict+Optimize is a recently proposed framework which combines machine learning and constrained optimization, tackling optimization problems that contain parameters that are unknown at solving time. The goal is to predict the unknown parameters and use the estimates to solve for an estimated optimal solution to the optimization problem. However, all prior works have focused on the case where unknown parameters appear only in the optimization objective and not the constraints, for the simple reason that if the constraints were not known exactly, the estimated optimal solution might not even be feasible under the true parameters. The contributions of this paper are two-fold. First, we propose a novel and practically relevant framework for the Predict+Optimize setting, but with unknown parameters in both the objective and the constraints. We introduce the notion of a correction function, and an additional penalty term in the loss function, modelling practical scenarios where an estimated optimal solution can be modified into a feasible solution after the true parameters are revealed, but at an additional cost. Second, we propose a corresponding algorithmic approach for our framework, which handles all packing and covering linear programs. Our approach is inspired by the prior work of Mandi and Guns, though with crucial modifications and re-derivations for our very different setting. Experimentation demonstrates the superior empirical performance of our method over classical approaches.

Deep learning models are state-of-the-art in compressive spectral imaging (CSI) recovery. These methods use a deep neural network (DNN) as an image generator to learn non-linear mapping from compressed measurements to the spectral image. For instance, the deep spectral prior approach uses a convolutional autoencoder network (CAE) in the optimization algorithm to recover the spectral image by using a non-linear representation. However, the CAE training is detached from the recovery problem, which does not guarantee optimal representation of the spectral images for the CSI problem. This work proposes a joint non-linear representation and recovery network (JR2net), linking the representation and recovery task into a single optimization problem. JR2net consists of an optimization-inspired network following an ADMM formulation that learns a non-linear low-dimensional representation and simultaneously performs the spectral image recovery, trained via the end-to-end approach. Experimental results show the superiority of the proposed method with improvements up to 2.57 dB in PSNR and performance around 2000 times faster than state-of-the-art methods.

We propose a novel Stochastic Frank-Wolfe (a.k.a. conditional gradient) algorithm for constrained smooth finite-sum minimization with a generalized linear prediction/structure. This class of problems includes empirical risk minimization with sparse, low-rank, or other structured constraints. The proposed method is simple to implement, does not require step-size tuning, and has a constant per-iteration cost that is independent of the dataset size. Furthermore, as a byproduct of the method we obtain a stochastic estimator of the Frank-Wolfe gap that can be used as a stopping criterion. Depending on the setting, the proposed method matches or improves on the best computational guarantees for Stochastic Frank-Wolfe algorithms. Benchmarks on several datasets highlight different regimes in which the proposed method exhibits a faster empirical convergence than related methods. Finally, we provide an implementation of all considered methods in an open-source package.

Unmanned surface vessels (USVs) are widely used in ocean exploration and environmental protection fields. To ensure that USV can successfully perform its mission, trajectory planning and motion tracking are the two most critical technologies. In this paper, we propose a novel trajectory generation and tracking method for USV based on optimization theory. Specifically, the USV dynamic model is described with differential flatness, so that the trajectory can be generated by dynamic RRT* in a linear invariant system expression form under the objective of optimal boundary value. To reduce the sample number and improve efficiency, we adjust the trajectory through local optimization. The dynamic constraints are considered in the optimization process so that the generated trajectory conforms to the kinematic characteristics of the under-actuated hull, and makes it easier to be tracked. Finally, motion tracking is added with model predictive control under a sequential quadratic programming problem. Experimental results show the planned trajectory is more in line with the kinematic characteristics of USV, and the tracking accuracy remains a higher level.

We propose a two-phase systematical framework for approximation algorithm design and analysis via Lyapunov function. The first phase consists of using Lyapunov function as an input and outputs a continuous-time approximation algorithm with a provable approximation ratio. The second phase then converts this continuous-time algorithm to a discrete-time algorithm with almost the same approximation ratio along with provable time complexity. One distinctive feature of our framework is that we only need to know the parametric form of the Lyapunov function whose complete specification will not be decided until the end of the first phase by maximizing the approximation ratio of the continuous-time algorithm. Some immediate benefits of the Lyapunov function approach include: (i) unifying many existing algorithms; (ii) providing a guideline to design and analyze new algorithms; and (iii) offering new perspectives to potentially improve existing algorithms. We use various submodular maximization problems as running examples to illustrate our framework.

Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\mathcal{M}$. Algorithms designed in this framework usually require some geometrical description of the manifold, which typically includes tangent spaces, retractions, and gradients of the cost function. However, in many cases, only a subset (or none at all) of these elements can be accessed due to lack of information or intractability. In this paper, we propose a novel approach that can perform approximate Riemannian optimization in such cases, where the constraining manifold is a submanifold of $\R^{D}$. At the bare minimum, our method requires only a noiseless sample set of the cost function $(\x_{i}, y_{i})\in {\mathcal{M}} \times \mathbb{R}$ and the intrinsic dimension of the manifold $\mathcal{M}$. Using the samples, and utilizing the Manifold-MLS framework (Sober and Levin 2020), we construct approximations of the missing components entertaining provable guarantees and analyze their computational costs. In case some of the components are given analytically (e.g., if the cost function and its gradient are given explicitly, or if the tangent spaces can be computed), the algorithm can be easily adapted to use the accurate expressions instead of the approximations. We analyze the global convergence of Riemannian gradient-based methods using our approach, and we demonstrate empirically the strength of this method, together with a conjugate-gradients type method based upon similar principles.