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Explore constrained optimization through this 23-minute conference talk from JuliaCon Global 2025 that demonstrates how Interior-point Methods (IPMs) efficiently solve complex mathematical optimization problems across scientific and engineering domains. Learn how ConicSolve.jl, a Julia package, implements these methods to handle various problem classes including Linear Programming (LP), Quadratic Programming (QP), Second Order Cone Programming (SOCP), and Semidefinite Programming (SDP). Discover practical applications in robotics for collision-free trajectory computation, image processing for MRI denoising through matrix completion, and network design for maximizing data throughput with capacity constraints. Understand key challenges in solving large-scale constrained optimization problems with thousands of constraints and examine how array manipulation techniques and thoughtful API design decisions in ConicSolve.jl simplify the process for practitioners. Investigate strategies for exploiting problem structure and sparsity to enhance solver performance through practical examples including image denoising and max flow min cut problems. Gain comprehensive understanding of the optimization modeling process and acquire tools to tackle common challenges in constrained optimization problem solving using Conic IPMs.
