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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…
This is a tutorial and survey paper on Karush-Kuhn-Tucker (KKT) conditions, first-order and second-order numerical optimization, and distributed optimization. After a brief review of history of optimization, we start with some preliminaries…
We propose an extended variant of the reformulation and decomposition algorithm for solving a special class of mixed-integer bilevel linear programs (MIBLPs) where continuous and integer variables are involved in both upper- and lower-level…
Many optimization problems require hyperparameters, i.e., parameters that must be pre-specified in advance, such as regularization parameters and parametric regularizers in variational regularization methods for inverse problems, and…
Clustering is a hard discrete optimization problem. Nonconvex approaches such as low-rank semidefinite programming (SDP) have recently demonstrated promising statistical and local algorithmic guarantees for cluster recovery. Due to the…
This paper proposes an iterative method to solve Mixed-Integer Optimal Control Problems arising from systems with switched dynamics. The so-called relaxed problem plays a central role within this context. Through a numerical example, it is…
A new topology optimization method called the Proportional Topology Optimization (PTO) is presented. As a non-gradient method, PTO is simple to understand, easy to implement, and is also efficient and accurate at the same time. It is…
Bilevel programming has recently received attention in the literature due to its wide range of applications, including reinforcement learning and hyper-parameter optimization. However, it is widely assumed that the underlying bilevel…
We consider pessimistic bilevel stochastic programs in which the follower maximizes over a fixed compact convex set a strictly convex quadratic function, whose Hessian depends on the leader's decision. The resulting random variable is…
This paper is concerned with an optimization problem governed by the Kantorovich optimal transportation problem. This gives rise to a bilevel optimization problem, which can be reformulated as a mathematical problem with complementarity…
Boolean quadratic optimization problems occur in a number of applications. Their mixed integer-continuous nature is challenging, since it is inherently NP-hard. For this motivation, semidefinite programming relaxations (SDR's) are proposed…
A standard approach to solving optimistic bilevel linear programs (BLPs) is to replace the lower-level problem with its Karush-Kuhn-Tucker (KKT) optimality conditions and reformulate the resulting complementarity constraints using auxiliary…
Many real-world problems, such as those with fairness constraints, involve complex expectation constraints and large datasets, necessitating the design of efficient stochastic methods to solve them. Most existing research focuses on cases…
In this paper, we propose the Bi-Sub-Gradient (Bi-SG) method, which is a generalization of the classical sub-gradient method to the setting of convex bi-level optimization problems. This is a first-order method that is very easy to…
Hyperparameter tuning is an important task of machine learning, which can be formulated as a bilevel program (BLP). However, most existing algorithms are not applicable for BLP with non-smooth lower-level problems. To address this, we…
Stochastic bilevel optimization finds widespread applications in machine learning, including meta-learning, hyperparameter optimization, and neural architecture search. To extend stochastic bilevel optimization to distributed data, several…
Penalty-based methods have become popular for solving bilevel optimization (BLO) problems, thanks to their effective first-order nature. However, they often require inner-loop iterations to solve the lower-level (LL) problem and small…
We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step…
Usually, bilevel optimization problems need to be transformed into single-level ones in order to derive optimality conditions and solution algorithms. Among the available approaches, the replacement of the lower-level problem by means of…
Bilevel optimization (BO) has recently gained prominence in many machine learning applications due to its ability to capture the nested structure inherent in these problems. Recently, many hypergradient methods have been proposed as…