Bilevel programs are optimization problems where some variables are solutions to optimization problems themselves, and they arise in a variety of control applications, including: control of vehicle traffic networks, inverse reinforcement learning and inverse optimization, and robust control for human-automation systems. This paper develops a duality-based approach to solving bilevel programs where the lower level problem is convex. Our approach is to use partial dualization to construct a new dual function that is differentiable, unlike the Lagrangian dual that is only directionally differentiable. We use our dual to define a duality-based reformulation of bilevel programs, prove equivalence of our reformulation with the original bilevel program, and then introduce regularization to ensure constraint qualification holds. These technical results about our new dual and regularized duality-based reformulation are used to provide theoretical justification for an algorithm we construct for solving bilevel programs with a convex lower level, and we conclude by demonstrating the efficacy of our algorithm by solving two practical instances of bilevel programs.
In this paper, we exploit the so-called value function reformulation of the bilevel optimization problem to develop duality results for the problem. Our approach builds on Fenchel-Lagrange-type duality to establish suitable results for the bilevel optimization problem. First, we overview some standard duality results to show that they are not applicable to our problem. Secondly, via the concept of partial calmness, we establish weak and strong duality results. In particular, Lagrange, Fenchel-Lagrange, and Toland-Fenchel- Lagrange duality concepts are investigated for this type of problems und…
In this paper, we present difference of convex algorithms for solving bilevel programs in which the upper level objective functions are difference of convex functions, and the lower level programs are fully convex. This nontrivial class of bilevel programs provides a powerful modelling framework for dealing with applications arising from hyperparameter selection in machine learning. Thanks to the full convexity of the lower level program, the value function of the lower level program turns out to be convex and hence the bilevel program can be reformulated as a difference of convex bilevel prog…
This paper considers a bilevel program, which has many applications in practice. To develop effective numerical algorithms, it is generally necessary to transform the bilevel program into a single-level optimization problem. The most popular approach is to replace the lower-level program by its KKT conditions and then the bilevel program can be reformulated as a mathematical program with equilibrium constraints (MPEC for short). However, since the MPEC does not satisfy the Mangasarian-Fromovitz constraint qualification at any feasible point, the well-developed nonlinear programming theory cann…
This paper proposes a Real-Time Market (RTM) platform for an aggregator and its corresponding prosumers to participate in the electricity wholesale market. The proposed energy market platform is modeled as a bilevel optimization problem where the aggregator and the prosumers are considered as self-interest agents. The current state-of-the-art Mathematical Programming with Equilibrium Constraints (MPEC) and Mixed-Integer Programming (MIP) based approaches to solve bilevel optimization problems are not satisfactory for real-time applications. The computation time for such approaches grows expone…
Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we establish an exact reformulation of chance constrained problems as a bilevel problems with convex lower-levels. We then derive a tractable penalty approach, where the penalized objective is a difference-of-convex function that we minimize with a suitable bundle algorithm. We release …
We consider a scalar objective minimization problem over the solution set of another optimization problem. This problem is known as simple bilevel optimization problem and has drawn a significant attention in the last few years. Our inner problem consists of minimizing the sum of smooth and nonsmooth functions while the outer one is the minimization of a smooth convex function. We propose and establish the convergence of a fixed-point iterative method with inertial extrapolation to solve the problem. Our numerical experiments show that the method proposed in this paper outperforms the currentl…
This paper introduces and studies the convergence properties of a new class of explicit $ε$-subgradient methods for the task of minimizing a convex function over the set of minimizers of another convex minimization problem. The general algorithm specializes to some important cases, such as first-order methods applied to a varying objective function, which have computationally cheap iterations. We present numerical experimentation regarding certain applications where the theoretical framework encompasses efficient algorithmic techniques, enabling the use of the resulting methods to solve very l…
Bilevel optimization formulates hierarchical decision-making processes that arise in many real-world applications such as in pricing, network design, and infrastructure defense planning. In this paper, we consider a class of bilevel optimization problems where the upper-level problem features some integer variables while the lower-level problem enjoys strong duality. We propose a dedicated Benders decomposition method for solving this class of bilevel problems, which decomposes the Benders subproblem into two more tractable, sequentially solvable problems that can be interpreted as the upper a…
A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-Hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by a be…
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 study a class of bilevel programming problem where the inner objective function is strongly convex. More specifically, under some mile assumptions on the partial derivatives of both inner and outer objective functions, we present an approximation algorithm for solving this class of problem and provide its finite-time convergence analysis under different convexity assumption on the outer objective function. We also present an accelerated variant of this method which improves the rate of convergence under convexity assumption. Furthermore, we generalize our results under stocha…
Bilevel optimization has been successfully applied to many important machine learning problems. Algorithms for solving bilevel optimization have been studied under various settings. In this paper, we study the nonconvex-strongly-convex bilevel optimization under a decentralized setting. We design decentralized algorithms for both deterministic and stochastic bilevel optimization problems. Moreover, we analyze the convergence rates of the proposed algorithms in difference scenarios including the case where data heterogeneity is observed across agents. Numerical experiments on both synthetic and…
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…
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and global minimum using a sequence of semidefinite programming (SDP) relaxations and provide convergence results for the methods. Our scheme for problems with a convex lower-level problem involves solving a transformed equivalent single-level problem by a sequence of SDP relaxations; whereas our approach for general problems involving a non-convex polynomial low…
In this paper we study constraint qualifications and optimality conditions for bilevel programming problems. We strive to derive checkable constraint qualifications in terms of problem data and applicable optimality conditions. For the bilevel program with convex lower level program we discuss drawbacks of reformulating a bilevel programming problem by the mathematical program with complementarity constraints and present a new sharp necessary optimality condition for the reformulation by the mathematical program with a generalized equation constraint. For the bilevel program with a nonconvex l…
Bilevel optimization problems involve two nested objectives, where an upper-level objective depends on a solution to a lower-level problem. When the latter is non-convex, multiple critical points may be present, leading to an ambiguous definition of the problem. In this paper, we introduce a key ingredient for resolving this ambiguity through the concept of a selection map which allows one to choose a particular solution to the lower-level problem. Using such maps, we define a class of hierarchical games between two agents that resolve the ambiguity in bilevel problems. This new class of games…
Bilevel Programming Approach for Image Restoration Problems with Automatically Hyperparameter Selection
School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）；Center for Applied Mathematics of Henan Province, Henan University（河南省应用数学中心，河南大学）；School of Nursing and Health, Henan University（河南大学护理与健康学院）；School of Software, Henan University（河南大学软件学院）
We study a class of bilevel convex optimization problems where the goal is to find the minimizer of an objective function in the upper level, among the set of all optimal solutions of an optimization problem in the lower level. A wide range of problems in convex optimization can be formulated using this class. An important example is the case where an optimization problem is ill-posed. In this paper, our interest lies in addressing the bilevel problems, where the lower level objective is given as a finite sum of separate nondifferentiable convex component functions. This is the case in a varie…
We consider bilevel linear problems, where some parameters are stochastic, and the leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader's outcome can be modeled by a random variable, which we evaluate based on some law-invariant convex risk measure. A qualitative stability result under perturbations of the underlying probability distribution is presented. Moreover, for the expectation, the expected excess, and the upper semideviation, we establish Lipschitz continuity as well as sufficient conditions for differentiability. Fin…
On solving integer bilevel optimization problems with a non-convex quadratic follower objective function using disjunctive cuts
Koç University（博阿兹奇大学）；University of Augsburg（奥格斯堡大学）；University of Michigan（密歇根大学）；Vienna University of Economics and Business（维也纳经济与商业大学）；Johannes Kepler University Linz（林茨约翰·开普勒大学）
Motivated by high-dimensional nonlinear optimization problems as well as ill-posed optimization problems arising in image processing, we consider a bilevel optimization model where we seek among the optimal solutions of the inner level problem, a solution that minimizes a secondary metric. Our goal is to address the high-dimensionality of the bilevel problem, and the nondifferentiability of the objective function. Minimal norm gradient, sequential averaging, and iterative regularization are some of the recent schemes developed for addressing the bilevel problem. But none of them address the hi…
Directional first order approach for a class of bilevel programs
School of Mathematics, Hunan University（湖南大学数学学院）；School of Artificial Intelligence, Southern University of Science and Technology（南方科技大学人工智能学院）；Department of Mathematics and Statistics, University of Victoria（维多利亚大学数学与统计系）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
A learning approach to selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized in the regularization parameters. Conditions which ensure the existence of solutions to the bilevel optimization problem of interest are derived, and these conditions are verified for two relevant examples. Difficulties arising from the possible lack of convexity of the lower level problems are discussed. Optimality conditions are given provided that a reason…
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）
In this paper, we focus on the nonconvex-strongly-convex bilevel optimization problem (BLO). In this BLO, the objective function of the upper-level problem is nonconvex and possibly nonsmooth, and the lower-level problem is smooth and strongly convex with respect to the underlying variable $y$. We show that the feasible region of BLO is a Riemannian manifold. Then we transform BLO to its corresponding unconstrained constraint dissolving problem (CDB), whose objective function is explicitly formulated from the objective functions in BLO. We prove that BLO is equivalent to the unconstrained opti…
We consider bilevel linear problems, where the right-hand side of the lower level problems is stochastic. The leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader's outcome can be modeled by a random variable, which gives rise to a broad spectrum of models involving coherent or convex risk measures and stochastic dominance constraints. We outline Lipschitzian properties, conditions for existence and optimality, as well as stability results. Moreover, for finite discrete distributions, we discuss the special structure of equiva…
A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks. In this paper, we consider two different formulations of the bilevel hierarchical clustering problem, a discrete optimization problem which can be shown to be NP-hard. Our approach is to reformulate the problem as a continuous optimization problem by making some relaxations on the discreteness conditions. Then Nesterov's smoothing technique and a numerical algorithm for minimizing differences of convex functions called the DCA are applied to cope with the nonsmoothness and nonconvexity of the prob…
Dalian University of Technology（大连理工大学）；Peng Cheng Laboratory（鹏城实验室）；The University of Hong Kong（香港大学）；University of Victoria（维多利亚大学）；Southern University of Science and Technology（南方科技大学）
This paper firstly proposes a convex bilevel optimization paradigm to formulate and optimize popular learning and vision problems in real-world scenarios. Different from conventional approaches, which directly design their iteration schemes based on given problem formulation, we introduce a task-oriented energy as our latent constraint which integrates richer task information. By explicitly re-characterizing the feasibility, we establish an efficient and flexible algorithmic framework to tackle convex models with both shrunken solution space and powerful auxiliary (based on domain knowledge an…
