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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…
This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…
We propose an optimization proxy in terms of iterative implicit gradient methods for solving constrained optimization problems with nonconvex loss functions. This framework can be applied to a broad range of machine learning settings,…
Low rank approximation is a commonly occurring problem in many computer vision and machine learning applications. There are two common ways of optimizing the resulting models. Either the set of matrices with a given rank can be explicitly…
We study the optimal behavior of a bidder in a real-time auction subject to the requirement that a specified collections of heterogeneous items be acquired within given time constraints. The problem facing this bidder is cast as a…
We consider convex optimization problems formulated using dynamic programming equations. Such problems can be solved using the Dual Dynamic Programming algorithm combined with the Level 1 cut selection strategy or the Territory algorithm to…
This paper proposes a new algorithm -- the \underline{S}ingle-timescale Do\underline{u}ble-momentum \underline{St}ochastic \underline{A}pprox\underline{i}matio\underline{n} (SUSTAIN) -- for tackling stochastic unconstrained bilevel…
One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization…
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…
This paper studies the static economic optimization problem of a system with a single aggregator and multiple prosumers in a Real-Time Balancing Market (RTBM). The aggregator, as the agent responsible for portfolio balancing, needs to…
Bilevel programs model sequential decision interactions between two sets of players and find wide applications in real-world complex systems. In this paper, we consider a bilevel mixed-integer linear program with binary tender, wherein the…
We develop a backward-in-time machine learning algorithm that uses a sequence of neural networks to solve optimal switching problems in energy production, where electricity and fossil fuel prices are subject to stochastic jumps. We then…
We consider joint optimization and learning problems arising in real-time decision systems. While most existing work focuses primarily on convex, revenue-based objectives, we extend this line of research to multi-objective formulations. In…
Existing decentralized stochastic optimization methods assume the lower-level loss function is strongly convex and the stochastic gradient noise has finite variance. These strong assumptions typically are not satisfied in real-world machine…
In this work, we develop analysis and algorithms for a class of (stochastic) bilevel optimization problems whose lower-level (LL) problem is strongly convex and linearly constrained. Most existing approaches for solving such problems rely…
Practical optimization problems may contain different kinds of difficulties that are often not tractable if one relies on a particular optimization method. Different optimization approaches offer different strengths that are good at…
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…
Topology optimization problems often support multiple local minima due to a lack of convexity. Typically, gradient-based techniques combined with continuation in model parameters are used to promote convergence to more optimal solutions;…
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…
Finding optimal bidding strategies for generation units in electricity markets would result in higher profit. However, it is a challenging problem due to the system uncertainty which is due to the unknown other generation units' strategies.…