Bilevel optimization is an important class of optimization problems where one optimization problem is nested within another. While various methods have emerged to address unconstrained general bilevel optimization problems, there has been a noticeable gap in research when it comes to methods tailored for the constrained scenario. The few methods that do accommodate constrained problems, often exhibit slow convergence rates or demand a high computational cost per iteration. To tackle this issue, our paper introduces a novel single-loop projection-free method employing a nested approximation technique. This innovative approach not only boasts an improved per-iteration complexity compared to existing methods but also achieves optimal convergence rate guarantees that match the best-known complexity of projection-free algorithms for solving convex constrained single-level optimization problems. In particular, when the hyper-objective function corresponding to the bilevel problem is convex, our method requires $\tilde{\mathcal{O}}(ε^{-1})$ iterations to find an $ε$-optimal solution. Moreover, when the hyper-objective function is non-convex, our method's complexity for finding an $ε$-stationary point is $\mathcal{O}(ε^{-2})$. To showcase the effectiveness of our approach, we present a series of numerical experiments that highlight its superior performance relative to state-of-the-art methods.
Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation
Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳应用数学国家研究中心）；City University of Hong Kong（香港城市大学）
Bilevel Learning with Inexact Stochastic Gradients
Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；Department of Electronics & Electrical Communication Engineering, Indian Institute of Technology (IIT) Kharagpur（印度理工学院Kharagpur电子与电气通信工程系）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学学院）
Derivative-Free Bilevel Optimization with Inexact Lower-Level Solutions
Dipartimento di Ingegneria Informatica, Automatica e Gestionale “A. Ruberti”, “Sapienza” Università di Roma（罗马大学信息与自动化与管理工程学院）；Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, Consiglio Nazionale delle Ricerche（意大利国家研究委员会系统与信息技术分析研究所）
In this work, the demand Adjustment Problem (DAP) associated to urban traffic planning is studied. The framework for the formulation of the DAP is mathematical programming with equilibrium constraints. In particular, if the optimization program associated to the equilibrium constraints is considered, the DAP results in a bilevel optimization problem. In this approach the DAP via the Inexact Restoration method is treated.
An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learning
EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa)（EPSRC统计应用数学博士培训中心（SAMBa））；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）；Department of Electronics & Electrical Communication Engineering, Indian Institute of Technology (IIT), Kharagpur（印度理工学院（Kharagpur）电子与电气通信工程系）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学学院）
University of Bath（巴斯大学）；Australian National University（澳大利亚国立大学）
Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. It is common when solving the upper-level problem to assume access to exact solutions of the lower-level problem, which is practically infe…
In this paper, we propose a globally convergent method for solving constrained nonlinear systems. The method combines an efficient Newton conditional gradient method with a derivative-free and nonmonotone linesearch strategy. The global convergence analysis of the proposed method is established under suitable conditions, and some preliminary numerical experiments are given to illustrate its performance.
In this paper, we propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate are established by using a general majorant condition. Two applications of such condition are provided: one is for functions whose the derivative satisfies Holder-like condition and the other is for functions that satisfies a Smale condition, which includes a substantial class of analytic functions. Some preliminaries numerical experiments illustrating the applicability of the proposed method …
Bilevel Learning via Inexact Stochastic Gradient Descent
EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath（巴斯大学统计应用数学博士培训中心）；Australian Research Council（澳大利亚研究理事会）；Indian Institute of Technology (IIT), Kharagpur（印度理工学院（Kharagpur分校））；University of Melbourne（墨尔本大学）；Department of Electronics & Electrical Communication Engineering（电子与电气通信工程系）；School of Mathematics and Statistics（数学与统计学学院）
University of Waterloo（滑铁卢大学）；University of Victoria（维多利亚大学）；Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）
Gradient-based optimization methods for hyperparameter tuning guarantee theoretical convergence to stationary solutions when for fixed upper-level variable values, the lower level of the bilevel program is strongly convex (LLSC) and smooth (LLS). This condition is not satisfied for bilevel programs arising from tuning hyperparameters in many machine learning algorithms. In this work, we develop a sequentially convergent Value Function based Difference-of-Convex Algorithm with inexactness (VF-iDCA). We show that this algorithm achieves stationary solutions without LLSC and LLS assumptions for b…
We study a class of algorithms for solving bilevel optimization problems in both stochastic and deterministic settings when the inner-level objective is strongly convex. Specifically, we consider algorithms based on inexact implicit differentiation and we exploit a warm-start strategy to amortize the estimation of the exact gradient. We then introduce a unified theoretical framework inspired by the study of singularly perturbed systems (Habets, 1974) to analyze such amortized algorithms. By using this framework, our analysis shows these algorithms to match the computational complexity of oracl…
The classical convex feasibility problem in a finite dimensional Euclidean space is studied in the present paper. We are interested in two cases. First, we assume to know how to compute an exact project onto one of the sets involved and the other set is compact such that the conditional gradient (CondG) method can be used for computing efficiently an inexact projection on it. Second, we assume that both sets involved are compact such that the CondG method can be used for computing efficiently inexact projections on them. We combine alternating projection method with CondG method to design a ne…
We suggest simple modifications of the conditional gradient method for smooth optimization problems, which maintain the basic convergence properties, but reduce the implementation cost of each iteration essentially. Namely, we propose the step-size procedure without any line-search, and inexact solution of the direction finding subproblem. Preliminary results of computational tests confirm efficiency of the proposed modifications.
Moreau Envelope Based Difference-of-weakly-Convex Reformulation and Algorithm for Bilevel Programs
University of British Columbia（不列颠哥伦比亚大学）；University of Victoria（维多利亚大学）；Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）；Peng Cheng Laboratory（鹏城实验室）
This paper studies mean-risk portfolio optimization models using the conditional value-at-risk (CVaR) as a risk measure. We also employ a cardinality constraint for limiting the number of invested assets. Solving such a cardinality-constrained mean-CVaR model is computationally challenging for two main reasons. First, this model is formulated as a mixed-integer optimization (MIO) problem because of the cardinality constraint, so solving it exactly is very hard when the number of investable assets is large. Second, the problem size depends on the number of asset return scenarios, and the comput…
In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in the authors' previous paper, which we denote ICGALP, that allows for errors in the computation of several important quantities. In particular this allows one to compute some gradients, proximal terms, and/or linear minimization oracles in an inexact fashion that facilitates the practical application of the algorithm to computationally intensive settings, e.g. in high (or possibly infinite) dimensional Hilbert spaces commonly found in machine learning problems. The algorithm is able to solve…
Simple bilevel problems are optimization problems in which we want to find an optimal solution to an inner problem that minimizes an outer objective function. Such problems appear in many machine learning and signal processing applications as a way to eliminate undesirable solutions. %However, since these problems do not satisfy regularity conditions, they are often hard to solve exactly and are usually solved via iterative regularization. In the past few years, several algorithms were proposed to solve these bilevel problems directly and provide a rate for obtaining feasibility, assuming that…
In this paper, we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems, and variational inequalities. This framework allows obtaining many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing kno…
A Decomposed Bilevel Search for Variable-Metric Proximal Gradient Methods
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院、中国科学院大学）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we investigate the iteration complexity and convergence rates of various subgradient methods for solving quasi-convex optimization in a unified framework. In particular, we consider a sequence satisfying a general (inexact) basic inequality, and investigate the global convergence theorem and the iteration complexity when using the constant, diminishing or dynamic stepsi…
This paper studies distributionally robust chance constrained programs (DRCCPs), where the uncertain constraints must be satisfied with at least a probability of a prespecified threshold for all probability distributions from the Wasserstein ambiguity set. As DRCCPs are often nonconvex and challenging to solve optimally, researchers have been developing various convex inner approximations. Recently, ALSO-X has been proven to outperform the conditional value-at-risk (CVaR) approximation of a regular chance constrained program when the deterministic set is convex. In this work, we relax this ass…
We develop subgradient- and gradient-based methods for minimizing strongly convex functions under a notion which generalizes the standard Euclidean strong convexity. We propose a unifying framework for subgradient methods which yields two kinds of methods, namely, the Proximal Gradient Method (PGM) and the Conditional Gradient Method (CGM), unifying several existing methods. The unifying framework provides tools to analyze the convergence of PGMs and CGMs for non-smooth, (weakly) smooth, and further for structured problems such as the inexact oracle models. The proposed subgradient methods yie…
We provide a framework for computing the exact worst-case performance of any algorithm belonging to a broad class of oracle-based first-order methods for composite convex optimization, including those performing explicit, projected, proximal, conditional and inexact (sub)gradient steps. We simultaneously obtain tight worst-case guarantees and explicit instances of optimization problems on which the algorithm reaches this worst-case. We achieve this by reducing the computation of the worst-case to solving a convex semidefinite program, generalizing previous works on performance estimation by Dr…
In this paper, we propose a new algorithm combining the Douglas-Rachford (DR) algorithm and the Frank-Wolfe algorithm, also known as the conditional gradient (CondG) method, for solving the classic convex feasibility problem. Within the algorithm, which will be named {\it Approximate Douglas-Rachford (ApDR) algorithm}, the CondG method is used as a subroutine to compute feasible inexact projections on the sets under consideration, and the ApDR iteration is defined based on the DR iteration. The ApDR algorithm generates two sequences, the main sequence, based on the DR iteration, and its corres…
