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1. IPO: Interior-point Policy Optimization under Constraints

   https://doi.org/10.1609/aaai.v34i04.5932  

   Liu, Yongshuai; Ding, Jiaxin; Liu, Xin (January 2020, Proceedings of the AAAI Conference on Artificial Intelligence)

   In this paper, we study reinforcement learning (RL) algorithms to solve real-world decision problems with the objective of maximizing the long-term reward as well as satisfying cumulative constraints. We propose a novel first-order policy optimization method, Interior-point Policy Optimization (IPO), which augments the objective with logarithmic barrier functions, inspired by the interior-point method. Our proposed method is easy to implement with performance guarantees and can handle general types of cumulative multiconstraint settings. We conduct extensive evaluations to compare our approach with state-of-the-art baselines. Our algorithm outperforms the baseline algorithms, in terms of reward maximization and constraint satisfaction. 

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2. An Inexact Variable Metric Proximal Point Algorithm for Generic Quasi-Newton Acceleration

   https://doi.org/https://epubs.siam.org/doi/10.1137/17M1125157  

   Lin, Hongzhou; Mairal, Julien; Harchaoui, Zaid (January 2019, SIAM journal on optimization)

   We propose an inexact variable-metric proximal point algorithm to accelerate gradient-based optimization algorithms. The proposed scheme, called QNing can be notably applied to incremental first-order methods such as the stochastic variance-reduced gradient descent algorithm (SVRG) and other randomized incremental optimization algorithms. QNing is also compatible with composite objectives, meaning that it has the ability to provide exactly sparse solutions when the objective involves a sparsity-inducing regularization. When combined with limited-memory BFGS rules, QNing is particularly effective to solve high-dimensional optimization problems, while enjoying a worst-case linear convergence rate for strongly convex problems. We present experimental results where QNing gives significant improvements over competing methods for training machine learning methods on large samples and in high dimensions. 

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3. Confidence Intervals for Projections of Partially Identified Parameters

   https://doi.org/10.3982/ECTA14075  

   Kaido, Hiroaki; Molinari, Francesca; Stoye, Jörg (January 2019, Econometrica)

   We propose a bootstrap‐based calibrated projection procedure to build confidence intervals for single components and for smooth functions of a partially identified parameter vector in moment (in)equality models. The method controls asymptotic coverage uniformly over a large class of data generating processes. The extreme points of the calibrated projection confidence interval are obtained by extremizing the value of the function of interest subject to a proper relaxation of studentized sample analogs of the moment (in)equality conditions. The degree of relaxation, or critical level, is calibrated so that the function of θ , not θ itself, is uniformly asymptotically covered with prespecified probability. This calibration is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive. Nonetheless, the program defining an extreme point of the confidence interval is generally nonlinear and potentially intricate. We provide an algorithm, based on the response surface method for global optimization, that approximates the solution rapidly and accurately, and we establish its rate of convergence. The algorithm is of independent interest for optimization problems with simple objectives and complicated constraints. An empirical application estimating an entry game illustrates the usefulness of the method. Monte Carlo simulations confirm the accuracy of the solution algorithm, the good statistical as well as computational performance of calibrated projection (including in comparison to other methods), and the algorithm's potential to greatly accelerate computation of other confidence intervals. 

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4. Adaptive and Universal Algorithms for Variational Inequalities with Optimal Convergence

   https://doi.org/10.1609/aaai.v36i6.20609  

   Ene, Alina; Nguyen, Huy L. (February 2022, AAAI Conference on Artificial Intelligence)

   We develop new adaptive algorithms for variational inequalities with monotone operators, which capture many problems of interest, notably convex optimization and convex-concave saddle point problems. Our algorithms automatically adapt to unknown problem parameters such as the smoothness and the norm of the operator, and the variance of the stochastic evaluation oracle. We show that our algorithms are universal and simultaneously achieve the optimal convergence rates in the non-smooth, smooth, and stochastic settings. The convergence guarantees of our algorithms improve over existing adaptive methods and match the optimal non-adaptive algorithms. Additionally, prior works require that the optimization domain is bounded. In this work, we remove this restriction and give algorithms for unbounded domains that are adaptive and universal. Our general proof techniques can be used for many variants of the algorithm using one or two operator evaluations per iteration. The classical methods based on the ExtraGradient/MirrorProx algorithm require two operator evaluations per iteration, which is the dominant factor in the running time in many settings. 

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5. An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization

   https://doi.org/10.3390/e25020330  

   Wu, Zeguan; Mohammadisiahroudi, Mohammadhossein; Augustino, Brandon; Yang, Xiu; Terlaky, Tamás (February 2023, Entropy)

   Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. Interior point methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization problems. IPMs solve a Newton linear system at each iteration to compute the search direction; thus, QLSAs can potentially speed up IPMs. Due to the noise in contemporary quantum computers, quantum-assisted IPMs (QIPMs) only admit an inexact solution to the Newton linear system. Typically, an inexact search direction leads to an infeasible solution, so, to overcome this, we propose an inexact-feasible QIPM (IF-QIPM) for solving linearly constrained quadratic optimization problems. We also apply the algorithm to ℓ1-norm soft margin support vector machine (SVM) problems, and demonstrate that our algorithm enjoys a speedup in the dimension over existing approaches. This complexity bound is better than any existing classical or quantum algorithm that produces a classical solution. 

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