The mathematical abstractions chosen to represent nonlinear optimization problems can influence the algorithms and software used for solving and modeling the problem. Graph theory is an abstraction that can be applied to a variety of nonlinear optimization problems, where nodes of a graph can contain objective functions, variables, and constraints and where edges contain constraints for variables on two or more nodes. Representing optimization problems as graphs enables unique algorithm implementations, allows for the identification of properties specific to the graph, and can be performed using open-source software. In this talk, we present how nonlinear optimization problems can be constructed using the open-source Julia package Plasmo.jl, we present different algorithms and solution schemes that can be applied to these graph-based problems, and we discuss properties specific to the graph-based representation. Overall, the graph-based representation can simplify the modeling of optimization problems and provides a framework for implementing algorithms that exploit the graph structure.

Derivative-based iterative methods for nonlinearly constrained nonconvex optimization usually share common algorithmic components, such as strategies for computing a descent direction and mechanisms that promote global convergence. Based on this observation, we introduce an abstract framework with four common ingredients that describes most derivative-based iterative methods and unifies their workflows. We then present Uno, a modular C++ solver that implements our abstract framework and allows the automatic generation of various strategy combinations with no programming effort from the user. Uno is meant to
(1) organize mathematical optimization strategies into a coherent hierarchy;
(2) offer a wide range of efficient and robust methods that can be compared for a given instance;
(3) reduce the cost of development and maintenance of multiple optimization solvers; and
(4) enable researchers to experiment with novel optimization strategies while leveraging established subproblem solvers and interfaces to modeling languages.
We demonstrate that Uno is highly competitive against state-of-the-art solvers such as filterSQP, IPOPT, SNOPT, MINOS, LANCELOT, LOQO and CONOPT on a set of 429 problems from the CUTEst collection.
Uno is available as open-source software under the MIT license at https://github.com/cvanaret/Uno.

This paper introduces a framework for solving alternating current optimal power flow (ACOPF) problems using graphics processing units (GPUs). While GPUs have demonstrated remarkable performance in various computing domains, their application in ACOPF has been limited due to challenges associated with porting sparse automatic differentiation (AD) and sparse linear solver routines to GPUs. We address these issues with two key strategies. First, we utilize a single-instruction, multiple-data abstraction of nonlinear programs. This approach enables the specification of model equations while preserving their parallelizable structure and, in turn, facilitates the parallel AD implementation. Second, we employ a condensed-space interior-point method (IPM) with an inequality relaxation. This technique involves condensing the Karush--Kuhn--Tucker (KKT) system into a positive definite system. This strategy offers the key advantage of being able to factorize the KKT matrix without numerical pivoting, which has hampered the parallelization of the IPM algorithm. By combining these strategies, we can perform the majority of operations on GPUs while keeping the data residing in the device memory only. Comprehensive numerical benchmark results showcase the advantage of our approach. Remarkably, our implementations -- MadNLP.jl and ExaModels.jl -- running on NVIDIA GPUs achieve an order of magnitude speedup compared with state-of-the-art tools running on contemporary CPUs.

We present our current developments of SQP-type solvers for direct optimal control within the acados software framework which is a versatile software framework that implements fast solvers exploiting the specific optimal control problem structure. While established solvers within acados mainly focus on full-step algorithms to achieve fast execution times, this talk focuses on recent work towards a fast and reliable SQP algorithm. The presented approach combines a funnel to drive global convergence with an l1-relaxation of the constraints to avoid infeasible subproblems. The SQP solver is planned to be combined with a DDP-based feasibility solver in a novel two-phase feasibility framework. Rather than achieving feasibility with respect to all iterations, a feasible iterate is aimed to be found after a given time window. The novel algorithm first solves for an optimal solution and eventually switches to a feasibility phase ensuring a feasible iterate at the end of the time window. Simulation results on time-optimal control in motion planning will be presented.