AMG with Filtering An Efficient Preconditioner for Large Scale Contact Mechanics Interior Point Optimization
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Mathematics and Computing↗
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Quantum Interior Point Methods for Semidefinite Optimization
We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem n , but has worse dependence on other numerical parameters.
Augustino, Brandon↗
Optimization to Generate Equations of State for Hydrogen Production
On a high level, the larger project in question, HydroGEN, aims to develop software used for finding equations of state (EOS) to optimize catalyst configuration for H 2 production through water splitting. In particular, this summer project focused on solving the nonlinear equations used in fitting the equations. This problem involved using Python to solve a linear system with nonlinear constraints. In order for this to be achieved, Pyomo was used to build a model and the solver Ipopt, interior point optimizer, was used. Pyomo is a Python-based language developed at Sandia; it is an optimization modeling language. Rather than solving the entire problem at once, a toy problem was created, simplifying the problem down to the most important focus. This problem had a known solution, comparable to the calculated solution to assess accuracy and as progress was made towards finding solutions, complexity was gradually added to the problem. After building and solving the toy problem, it was found that it gave reasonably accurate solutions, better compared to the two existing solvers previously used with this project in terms of functionality. The solver is now ready for implementation into the project’s main software.
08 HYDROGEN↗
A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints
Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.
PDE-constrained optimization↗
Symmetric Random Butterfly Transform (SRBT) Based Preconditioner
Summary of work using Symmetric Random Butterfly Transformation (SRBT) in conjunction with Incomplete LDL T factorization as a preconditioner for FGMRES solver as a way of solving linear systems arising from interior point methods applied to power system problems. These linear systems have proven difficult to parallelize and this represents a possible route forward.
interior point optimization↗
Algebraic Multigrid with Filtering: An Efficient Preconditioner for Interior Point Methods in Large-Scale Contact Mechanics Optimization
Large-scale contact mechanics simulations are crucial in many engineering fields such as structural design and manufacturing. In the frictionless case, contact can be modeled by minimizing an energy functional; however, these problems are often nonlinear, nonconvex, and increasingly difficult to solve as mesh resolution increases. In this work, we employ a Newton-based interior-point (IP) filter line-search method, an effective approach for large-scale constrained optimization. While this method converges rapidly, each iteration requires solving a large saddle-point linear system that becomes ill-conditioned as the optimization process converges, largely due to IP treatment of the contact constraints. Such ill-conditioning can hinder solver scalability and increase iteration counts with mesh refinement. Here, to address this, we introduce a novel preconditioner, algebraic multigrid with filtering (AMGF), tailored to the Schur complement of the saddle-point system. Building on the classical AMG solver, commonly used for elasticity, we augment it with a specialized subspace correction that filters near null space components introduced by contact interface constraints. Through theoretical analysis and numerical experiments on a range of linear and nonlinear contact problems, we demonstrate that the proposed solver achieves mesh independent convergence and maintains robustness against the ill-conditioning that notoriously plagues IP methods. These results indicate that AMGF makes contact mechanics simulations more tractable and broadens the applicability of Newton-based IP methods in challenging engineering scenarios. More broadly, AMGF is well suited for problems, optimization or otherwise, where solver performance is limited by a low-dimensional subspace, such as those arising from localized constraints, interface conditions, or model heterogeneities. This makes the method widely applicable beyond contact mechanics and constrained optimization.
Mathematics and Computing↗
Implementing a unified solver for nonlinearly constrained optimization
SQP and interior-point methods (also referred to as Lagrange-Newton methods) typically share key algorithmic components, such as strategies for computing descent directions and mechanisms that promote global convergence. Building on this insight, we introduce a unifying framework with eight building blocks that abstracts the workflows of Lagrange-Newton methods. We then present Uno, a modular C++ solver that implements our unifying framework and allows the automatic combination of a wide range of strategies 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) enable researchers to experiment with novel optimization strategies; and (4) reduce the cost of development and maintenance of multiple optimization solvers. Uno’s software design allows user to compose new customized solvers for emerging optimization areas such as robust optimization or optimization problems with complementarity constraints, while building on reliable nonlinear optimization techniques. We demonstrate that Uno is highly competitive against state-of-the-art solvers filterSQP, IPOPT, SNOPT, MINOS, LANCELOT, LOQO, and CONOPT on a subset of 429 small problems from the CUTE collection. Uno is available as open-source software under the MIT license at https://github.com/cvanaret/Uno and via its C, Julia, Python, Fortran, and AMPL interfaces.
97 MATHEMATICS AND COMPUTING↗
Riemannian Optimization Applied to AC Optimal Power Flow: Preprint
The nonlinear, nonconvex AC optimal power flow problem is of growing importance as the nature of the power grid evolves. This problem can be difficult to solve for interior point methods. However, the advent of optimization algorithms over smooth Riemannian manifolds presents an alternative approach. The nonlinear, nonconvex constraints in the AC power flow problem form an embedded submanifold of Euclidean space. In this paper, the authors explore the performance of Riemannian optimization algorithms for the ACOPF problem where the optimization is performed directly on the AC power flow manifold. They demonstrate that these are viable computational alternatives to interior point methods. This is done by using Julia and the packages PowerModels.jl and Manopt.jl.
manifold optimization↗
Riemannian Optimization Applied to AC Optimal Power Flow
The nonlinear, nonconvex AC optimal power flow problem is of growing importance as the nature of the power grid evolves. This problem can be difficult to solve for interior point methods. However, the advent of optimization algorithms over smooth Riemannian manifolds presents an alternative approach. The nonlinear, nonconvex constraints in the AC power flow problem form an embedded submanifold of Euclidean space. In this paper, the authors explore the performance of Riemannian optimization algorithms for the ACOPF problem where the optimization is performed directly on the AC power flow manifold. This is done by using the Julia programming language and the Julia packages PowerModels.jl and Manopt.jl.
AC optimal power flow↗
Exploiting Power Flow Manifold to Solve AC Optimal Power Flow
AC optimal power flow has proven difficult to solve with interior point methods on GPUs. This is largely due to challenging linear algebra problems that current state of the art massively parallel linear solvers struggle with. However, the advent of Riemannian optimization techniques and the fact that the power flow equations form a smooth manifold present an alternative approach. In this talk, we present the basics of Riemannian optimization techniques in which optimization is done directly on a manifold. Then we present computational results showing that Riemannian techniques are capable of producing solutions of comparable quality as interior point methods.
AC optimal power flow↗
An Iterative Approach for Solving the SCOPF Problem Applying LP, SOCP, and NLP Subproblems
We propose to develop efficient algorithms and software for the SCOPF problem. We will employ an iterative approach that will: a) use linear subproblems and other active set filtering techniques to identify the most important contingencies and drastically reduce the SCOPF model size; b) solve SOCP relaxations of the reduced SCOPF to converge to the neighborhood of the global optimal solution and establish a lower bound on the solution, and; c) use a non-convex, nonlinear interior-point solver, Artelys Knitro, to converge quickly to the optimal solution. To identify the most effective approach, we will experiment with several techniques to identify the tradeoffs between contingency subproblem complexity and fast solvability.
29 ENERGY PLANNING, POLICY, AND ECONOMY↗
Parallel interior-point solver for block-structured nonlinear programs on SIMD/GPU architectures
Here, we investigate how to port the standard interior-point method to new exascale architectures for block-structured nonlinear programs with state equations. Computationally, we decompose the interior-point algorithm into two successive operations: the evaluation of the derivatives and the solution of the associated Karush-Kuhn-Tucker (KKT) linear system. Our method accelerates both operations using two levels of parallelism. First, we distribute the computations on multiple processes using coarse parallelism. Second, each process uses SIMD/GPU accelerators locally to accelerate the operations using fine-grained parallelism. The KKT system is reduced by eliminating the inequalities and the state variables from the corresponding equations. We demonstrate our method's capability on the supercomputer Polaris, a testbed for the future exascale Aurora system. Each node is equipped with four GPUs, a setup amenable to our two-level approach. Our experiments on the stochastic optimal power flow problem show that the reduction method is 50x faster than the sparse linear solver HSL MA57 running in serial on the CPU, and 6x faster than Pardiso running in parallel on CPU on the same number of processes.
97 MATHEMATICS AND COMPUTING↗
Optimal Power Flow Derived Sparse Linear Solver Benchmarks
Due to the changing nature of the power grid, it is increasingly important to be able to solve a high-fidelity optimal power-flow models on large power networks. This high-fidelity problem, called AC Optimal Power Flow (ACOPF), is a nonlinear, nonconvex optimization problem. One of the few reliable ways of solving such a problem is interior point methods. These methods result in sparse linear systems where the coefficient matrix is symmetric, indefinite and nearly always ill-conditioned. As such, they are particularly challenging for sparse linear solvers and represent a considerable computational bottleneck in solving the ACOPF problem. In this paper, we introduce a repository of linear systems captured from ACOPF problems when solved by the open-source optimizer IPOPT. These matrices are meant to be used as a test suite for sparse linear solver development.
97 MATHEMATICS AND COMPUTING↗
Homotopy Solver
This software implements parallel versions of an interior-point solver, based on the publicly available ipopt solver. Here we have full control over the linear solver and our algorithm is fully parallel thus enabling scalability to large-scale optimization problems. This package also has a parallel implementation of a homotopy solver developed under the scalable methods for contact LDRD project 23-ERD-017. This solver is an mfem-based implementation of algorithm described in ``A filter trust-region Newton continuation method for nonlinear complementarity problems''. Cosmin G. Petra, Nai-Yuan Chiang, Jingyi Wang, Tucker Hartland, and Michael Puso (submitted), LLNL-JRNL-869761.
Hartland, Tucker [Lawrence Livermore National Labo↗
Using Filter Methods to Guide Convergence for ADMM, with Applications to Nonnegative Matrix Factorization Problems
Nonconvex, nonlinear optimization problems arise naturally in parameter fitting and machine learning. While augmented Lagrangian methods have demonstrated robust convergence for classes of these problems, their convergence for block updates has been relatively unexplored outside of the context of the alternating direction method of multipliers (ADMM). ADMM has seen extensive use in these applications, but may exhibit uncertain convergence behavior in many practical nonconvex settings, and struggles with general nonlinear constraints. In contrast, filter methods have proved effective in enforcing convergence for sequential quadratic programming methods and interior point methods with feasibility criteria. We develop an ADMM-filter method for highly nonlinear and nonconvex problems. Here, we show convergence under mild assumptions for several types of coordinate descent schemes, and demonstrate our algorithm on nonnegative matrix factorization and completion problems in imaging and chemical spectrum analysis.
Nonconvex optimization↗
Implementation and validation of optimal start control strategy for air conditioners and heat pumps
Commercial buildings are responsible for approximately 20 % of the total energy consumption and greenhouse gas emissions in the United States. Over 85 % of these buildings lack building automation systems, and many are small (<50,000 square feet), underserved, and use rooftop units (RTUs) for heating, ventilation, and air-conditioning needs. Because these buildings lack proper energy management systems, several operational deficiencies lead to excess energy consumption. Studies have shown that managing the RTUs’ heating and cooling set points, schedules, setbacks, and optimal start can result in a 20 % to 25 % reduction in electricity consumption in small commercial buildings. These buildings typically use fixed schedules to start the RTUs 60 to 120 min before occupancy begins, which results in excess energy consumption. This paper presents research that demonstrates and evaluates the performance of four optimal start methods, which utilize data-based modeling as a key element in facilitating adaptive control in response to time-varying inputs while requiring minimal sensor inputs. The evaluation found energy savings in two commercial buildings equipped with RTUs by periodically alternating four different optimal start models during the cooling and heating season. The resulting energy savings are positive for all models and range from 2 to 5 kWh/day/unit. The units on the east side of the building showed higher savings, while interior units showed greater variability in savings due to the differences in capacities and room sizes. Savings were considerably greater during the heating season compared to the cooling season. The performance of all four models on Mondays was poor; models suggested a shorter optimal start time, which resulted in relatively larger errors. Finally, the future work will look at using a different model for the days after weekends and holidays.
42 ENGINEERING↗
Innovating the next generation of commercial smart building software
Nearly 30% of commercial building energy use is wasted due to equipment faults and HVAC controls problems. The result is increased emissions, compromised comfort and productivity, and less reliable coordination of building power needs with a clean grid. The energy impact alone represents $17 billion in potential savings. Today’s smart building software provides a robust solution to address these operational deficiencies. Energy management and information systems (EMIS) are saving up to 9% on average, with two-year paybacks. They are being incorporated into energy management processes, commissioning services, and utility programs. As effective as they are, two barriers prevent even deeper benefits; limited personnel to fix problems once they are identified, and the expense and time to manually implement changes in control systems. In partnership with the research community, the EMIS industry is developing new capabilities to overcome these barriers. Moving beyond siloed products for either fault detection and diagnostics, or optimal control, these new capabilities empower users to not only automatically identify faults, but also to push corrective action, and control improvements to their buildings. In this paper, several areas for enhancements are documented: ‘one-time’ correction of faults such as setpoints, schedules, and economizer lockouts; short-term active testing for automated proportional integral derivative (PID) loop tuning and functional testing; and continuous supervisory control for demand flexibility and year-round efficiency. Results are presented from a pair of partner implementations out of a dozen providers integrating these enhancements into their products, including field tests from across the country, and insights into operator acceptance and integration into operations and maintenance practices.
Casillas, Armando↗