We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.
Here, necessary optimal condition for Optimistic Bilevel programming problem is obtained in Asplund spaces. Also we have got necessary optimal conditions in finite dimensional spaces, by assuming differentiability on the given functions.
Bilevel optimization has been widely used in decision-making process. However, there still lacks an efficient algorithm to determine an optimal solution of a bilevel optimization problem, especially for a large-size problem. To bridge the gap, this paper proposes an efficient decomposition algorithm for a general bilevel linear programming(GBLP). The simulation results on large-size testing system demonstrate its correctness and efficiency.
This article continues our study on simple bilevel and simple MPEC problems. In this article we focus on developing algorithms. We show how using the idea of a gap function one can represent a simple MPEC as a simple bilevel problem with non-smooth data. This motivates us first to develop an algorithm for a simple bilevel problem with non-smooth data and modify the scheme for the same to develop an algorithm for the simple MPEC problem. We also discuss how the simple bilevel formulation of a simple MPEC can help us in formulating a stopping criteria for the simple MPEC problem
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…
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…
The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that t…
This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization probl…
In this paper we begin by discussing the simple bilevel programming problem (SBP) and its extension the simple mathematical programming problem under equilibrium constraints (SMPEC). Here we first define both these problems and study their interrelations. Next we study the various types of necessary and sufficient optimality conditions for the (SMPEC) problems; which occur under various reformulations. The optimality conditions for the (SBP) problem are special cases of the results obtained here when the lower level objective is the gradient of a convex function. Among the various optimality c…
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…
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）
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…
In this paper, we propose a combined approach with second-order optimality conditions of the lower level problem to study constraint qualifications and optimality conditions for bilevel programming problems. The new method is inspired by the combined approach developed by Ye and Zhu in 2010, where the authors combined the classical first-order and the value function approaches to derive new necessary optimality conditions. In our approach, we add a second-order optimality condition to the combined program as a new constraint. We show that when all known approaches fail, adding the second-order…
Bilevel Optimization Programming is used to model complex and conflicting interactions between agents, for example in Robust AI or Privacy-preserving AI. Integrating bilevel mathematical programming within deep learning is thus an essential objective for the Machine Learning community. Previously proposed approaches only consider single-level programming. In this paper, we extend existing single-level optimization programming approaches and thus propose Differentiating through Bilevel Optimization Programming (BiGrad) for end-to-end learning of models that use Bilevel Programming as a layer. B…
Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity-constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The…
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…
In this work we propose an improvement of the $x$-space algorithm developed for solving a class of min--max bilevel optimization problems (Tang Y., Richard J.P.P., Smith J.C. (2016), A class of algorithms for mixed-integer bilevel min--max optimization. Journal of Global Optimization, 66(2), 225--262). In this setting, the leader of the upper level problem aims at restricting the follower's decisions by minimizing an objective function, which the follower intends to maximize in the lower level problem by making decisions still available to her. The $x$-space algorithm solves upper and lower bo…
We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds a continuous wait-and-see decision after observing the leader's action and revelation of uncertainty. Only the information of the mean, covariance, and support is known. We formulate the problem as a distributionally robust (DR) two-stage problem. The pessimistic DR bilevel program is shown to be equivalent to a generic two-stage distributionally robust stochastic (nonlinear) program with both a random objective and ra…
A geometric reformulation of the bilevel parameter optimization problem to a single level non-linear programming problem with applications to phase equilibria
University of Bremen（不来梅大学）；Leibniz Institute for Materials Engineering - IWT（莱布尼兹材料工程研究所）
Shandong University（山东大学）；Renmin University of China（中国人民大学）；Tsinghua University（清华大学）；Huawei（华为）
The (gradient-based) bilevel programming framework is widely used in hyperparameter optimization and has achieved excellent performance empirically. Previous theoretical work mainly focuses on its optimization properties, while leaving the analysis on generalization largely open. This paper attempts to address the issue by presenting an expectation bound w.r.t. the validation set based on uniform stability. Our results can explain some mysterious behaviours of the bilevel programming in practice, for instance, overfitting to the validation set. We also present an expectation bound for the clas…
This paper studies, for the first time, a bilevel polynomial program whose constraints involve uncertain linear constraints and another uncertain linear optimization problem. In the case of box data uncertainty, we present a sum of squares polynomial characterization of a global solution of its robust counterpart where the constraints are enforced for all realizations of the uncertainties within the prescribed uncertainty sets. By characterizing a solution of the robust counterpart of the lower-level uncertain linear program under spectrahedral uncertainty using a new generalization of Farkas'…
The partial calmness for the bilevel programming problem (BLPP) is an important condition which ensures that a local optimal solution of BLPP is a local optimal solution of a partially penalized problem where the lower level optimality constraint is moved to the objective function and hence a weaker constraint qualification can be applied. In this paper we propose a sufficient condition in the form of a partial error bound condition which guarantees the partial calmness condition. We analyse the partial calmness for the combined program based on the Bouligand (B-) and the Fritz John (FJ) stati…
This paper studies bilevel polynomial optimization problems. To solve them, we give a method based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment-SOS relaxations are used to solve the polynomial optimizattion relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimizatio…
We study connections between optimistic bilevel programming problems and Generalized Nash Equilibrium Problems (GNEP)s. We remark that, when addressing bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a new GNEP model that is closely related to the bilevel program. We provide a complete analysis of the relationship between the "vertical" bilevel problem and the corresponding "horizontal" (one-level) GNEP model. We define classes of problems for which solutions of the bile…
The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an opti…
A mathematical programming problem with affine equilibrium constraints (AMPEC) is a bilevel programming problem where the lower one is a parametric affine variational inequality. We formulate some classes of bilevel programming in forms of MPEC. Then we use a regularization technique to formulate the resulting problem as a mathematical program with an additional constraint defined by the difference of two convex functions (DC function). A main feature of this DC decomposition is that the second component depends upon only the parameter in the lower problem. This property allows us to develop b…
A Penalty-Based Method for Communication-Efficient Decentralized Bilevel Programming
Amirkabir University of Technology（阿米尔卡比尔理工大学）；American University（美利坚大学）；University of Pennsylvania（宾夕法尼亚大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
The authors' paper in Optimization 63 (2014), 505-533, see Ref. [5], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analyzed in SIAM J. Optim. 22 (2012), 1309-1343; see Ref. [4], for the case of optimistic bilevel programs. One of the basic assumptions in both of these papers is that the functions involved in the problems are at least continuously differentiable. Motivated by the fact that many real-world applications of optimization involve func…
The emerging edge computing paradigm promises to provide low latency and ubiquitous computation to numerous mobile and Internet of Things (IoT) devices at the network edge. How to efficiently allocate geographically distributed heterogeneous edge resources to a variety of services is a challenging task. While this problem has been studied extensively in recent years, most of the previous work has largely ignored the preferences of the services when making edge resource allocation decisions. To this end, this paper introduces a novel bilevel optimization model, which explicitly takes the servic…
The bilevel program is an optimization problem where the constraint involves solutions to a parametric optimization problem. It is well-known that the value function reformulation provides an equivalent single-level optimization problem but it results in a nonsmooth optimization problem which never satisfies the usual constraint qualification such as the Mangasarian-Fromovitz constraint qualification (MFCQ). In this paper we show that even the first order sufficient condition for metric subregularity (which is in general weaker than MFCQ) fails at each feasible point of the bilevel program. We…
Farkas' Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of the alternative for integer programming and conic programming. We present theorems of the alternative for conic integer programming. We provide a nested procedure to construct a function that characterizes feasibility over right-hand sides and can determine which statement in a theorem of the alternative holds.
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…
On the role of semismoothness in the implicit programming approach to selected nonsmooth optimization problems
Institute of Information Theory and Automation, Czech Academy of Sciences（捷克科学院信息论与自动化研究所）；Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences（奥地利科学院约翰·拉登计算与应用数学研究所）；School of Mathematics, University of Birmingham（伯明翰大学数学学院）
We consider a degenerate nonsmooth and nonconvex optimization problem for which the standard constraint qualification such as the generalized Mangasarian Fromovitz constraint qualification (GMFCQ) may not hold. We use smoothing functions with the gradient consistency property to approximate the nonsmooth functions and introduce a smoothing sequential quadratic programming (SQP) algorithm under the exact penalty framework. We show that any accumulation point of a selected subsequence of the iteration sequence generated by the smoothing SQP algorithm is a Clarke stationary point, provided that t…
