Operation-Aware Deterministic Global Optimization of Carnot Battery Design using Hybrid Modeling

Categories Energy, Global Optimization Applications Tags , , , , ,

The global demand for grid-scale energy storage continues to increase. Carnot batteries (CBs) are not geographically constrained and consist of mature components. Furthermore, charging power, discharging power, and storage capacity can be sized independently and tailored to the intended use case. Designing an optimal CB also requires considering the resulting operational behavior. While design and … Read more

Advanced Geometrical Test for Interval Branch and Bound methods

Categories Global Optimization, Robust Optimization Tags , , , ,

The Interval Branch and Bound (IBB) method is a widely used approach for solving nonlinear programming problems, especially when a rigorous solution is required. It uses Interval Arithmetic to handle rounding errors. Although numerous variants of the IBB method have been proposed in the literature, relatively few implementations incorporate Karush-Kuhn-Tucker or Fritz-John (FJ) optimality conditions … Read more

De-risking solutions to optimization problems

Categories Bilevel Optimization, Convex and Nonsmooth Optimization, Robust Optimization Tags

We develop a cutting-plane methodology that adjusts solutions to optimization problems so as to reduce features that bring about exposure to risk, such as concentration of assets or resources. The methodology is agnostic to the representation of risk but has provably good attributes. Our procedure aims to reduce the appropriate risk metric without accruing a … Read more

Lower Bounds for Feasibility and Stationarity in First-Order Nonconvex Constrained Optimization

Categories Constrained Nonlinear Optimization, Nonlinear Optimization

We study oracle-complexity lower bounds for first-order methods applied to smooth equality-constrained nonconvex optimization, separating two components of approximate KKT accuracy: feasibility and stationarity. Under iteration-wise Jacobian regularity, we prove a lower bound of order \(\Omega(\frac{L_c\Delta_c}{\sigma^2}+\log\log(\frac{\sigma^2}{L_c\epsilon}))\) to achieve \(\epsilon\)-feasibility. For stationarity, we construct a nonlinear equality-constrained hard instance whose multiplier-minimized stationarity residual reduces exactly to … Read more

A Fletcher’s Augmented Lagrangian-Based Stochastic First-Order Method for Nonconvex Equality-Constrained Optimization

Categories Constrained Nonlinear Optimization, Nonlinear Optimization

In this paper, we study nonconvex equality-constrained optimization problems in which only stochastic first-order approximations of the objective and constraint functions are available. Owing to the stochasticity in both objective and constraints, most existing stochastic first-order methods incur relatively high oracle complexity, particularly in terms of stochastic constraint function evaluations. To address this issue, we … Read more

Constrained Variable Projection for Structured Problems

Categories Bilevel Optimization, Nonlinear Optimization, Optimization in Data Science Tags , ,

Variable projection is a classical technique for separable nonlinear least-squares problems, in which variables that enter linearly are eliminated exactly, yielding a reduced nonlinear problem. By expressing this framework as a particular instance of a broader class of bilevel optimization problems, we develop a constrained variable-projection framework for data-science models, where the remaining variables are … Read more

Neural Assortment Optimization

Categories Combinatorial Optimization, Nonsmooth Optimization, Yield Management

Assortment optimization selects a subset of items to maximize expected revenue under a discrete choice model and is widely used in revenue management and online platforms. Its combinatorial nature creates a practical tension among generality, scalability, and provable guarantees: model-specific algorithms can be strong when their structural assumptions hold, but are hard to adapt across … Read more

Optimal Macroitem Sequences in the Precedence Constrained Knapsack Problem

Categories Combinatorial Optimization, Graphs and Matroids, Linear Programming Tags , , ,

The Precedence Constrained Knapsack Problem (PCKP) asks for a maximum-profit subset of items, subject to a knapsack capacity constraint and precedence constraints encoded by a directed acyclic graph. We study the structure of optimal solutions of the Linear Programming (LP) relaxation of the natural Integer Linear Programming formulation of the PCKP. We introduce the notion … Read more

Local-to-Global Exactness of SDP Relaxations for Sparse QCQPs

Categories Global Optimization Theory, Quadratic Programming, Semi-definite Programming Tags , , , , , , , ,

We study exact semidefinite programming (SDP) relaxation for a given sparse quadratically constrained quadratic program (QCQP). The SDP relaxation is exact if, whenever it has an optimal solution, it admits a rank-at-most-one optimal solution that corresponds to an optimal solution of the QCQP. Using the maximal cliques of a chordal extension of the aggregate sparsity … Read more