Second order conditions of optimality for constrained optimization problems in finite dimensional spaces
Conditions of optimality for constrained optimization problem
Masserli, E. J.↗
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Second order conditions of optimality for constrained optimization problems in finite dimensional spaces.
Second order necessary conditions of optimality for constrained optimization problems in finite dimensional spaces with vanished cost function gradient at optimal point
Messerli, E. J.↗
Toward a scalable robust security-constrained optimal power flow using a proximal projection bundle method
Robust security-constrained optimal power flow (rSCOPF) aims to find the worst-case contingencies of alternating current optimal power flow (ACOPF) in power systems. With the rise of GPU architectures on the upcoming supercomputer architectures, optimization algorithms that rely on sparse linear algebra and indefinite linear systems are becoming increasingly hard to solve efficiently (e.g. interior-point method). To address this we revisit a maximin optimization formulation of the rSCOPF and the single-level mixed-integer semidefinite programming (MISDP) reformulation, which is obtained by taking the Lagrangian relaxation of the inner minimization ACOPF problem. In this paper, we focus on the development of a proximal projection bundle method (PPBM) for solving continuous relaxation node subproblems of the MISDP problem, based primarily on the well-known alternating direction method of multipliers. Cutting planes reminiscent of bundle method ideas are also applied in coordination with updates of the proximal parameter. The cutting-plane method can generate a large number of linear inequalities, leading to a large scale but decomposable quadratic programming (QP) subproblem that is amenable to GPUs. We present the numerical results on the IEEE 30, 57, 118, and 300-bus systems by using our PBMM method. We discuss the main computational bottleneck of our method, which is the time taken to solve each iteration of a QP subproblem instance of the PPBM, and how GPU architectures can accelerate this solution process.
bundle method↗
Machine learning models for PDE constrained optimization
Partial differential equation (PDE)-constrained optimization problems arise in a variety of scientific and engineering applications, such as topology optimization, electrodynamics, fluid dynamics, and structural dynamics. However, these problems are often challenging and computationally expensive to solve, due to the need to solve the PDEs within the optimization loop. One approach to reducing the computational cost of these methods while providing convergence guarantees is through inexact trust region methods; this method uses lower fidelity solutions of the PDE at early stages of the optimization and adjusts the required accuracy of inexact PDE solvers as the optimization progresses. In this work, we explore the use of machine learning based surrogate models with these inexact trust region methods. We first demonstrate the potential of this approach by using Gaussian processes as the surrogate model and test this on a simple PDE-constrained optimization problem. We then document explorations into improving the computational costs of evolutional deep neural network / neural Galerkin methods, with the eventual goal of using these methods with the inexact trust region algorithms. We are able to speed up these approaches, albeit at the cost of lower accuracy.
97 MATHEMATICS AND COMPUTING↗
Optimization with Neural Network Feasibility Surrogates: Formulations and Application to Security-Constrained Optimal Power Flow
In many areas of constrained optimization, representing all possible constraints that give rise to an accurate feasible region can be difficult and computationally prohibitive for online use. Satisfying feasibility constraints becomes more challenging in high-dimensional, non-convex regimes which are common in engineering applications. A prominent example that is explored in the manuscript is the security-constrained optimal power flow (SCOPF) problem, which minimizes power generation costs, while enforcing system feasibility under contingency failures in the transmission network. In its full form, this problem has been modeled as a nonlinear two-stage stochastic programming problem. In this work, we propose a hybrid structure that incorporates and takes advantage of both a high-fidelity physical model and fast machine learning surrogates. Neural network (NN) models have been shown to classify highly non-linear functions and can be trained offline but require large training sets. In this work, we present how model-guided sampling can efficiently create datasets that are highly informative to a NN classifier for non-convex functions. We show how the resultant NN surrogates can be integrated into a non-linear program as smooth, continuous functions to simultaneously optimize the objective function and enforce feasibility using existing non-linear solvers. Overall, this allows us to optimize instances of the SCOPF problem with an order of magnitude CPU improvement over existing methods.
24 POWER TRANSMISSION AND DISTRIBUTION↗
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↗
Constrained optimization for image restoration using nonlinear programming
The constrained optimization problem for image restoration, utilizing incomplete information and partial constraints, is formulated using nonlinear proramming techniques. This method restores a distorted image by optimizing a chosen object function subject to available constraints. The penalty function method of nonlinear programming is used. Both linear or nonlinear object function, and linear or nonlinear constraint functions can be incorporated in the formulation. This formulation provides a generalized approach to solve constrained optimization problems for image restoration. Experiments using this scheme have been performed. The results are compared with those obtained from other restoration methods and the comparative study is presented.
Yeh, C.-L.↗
Variable-Metric Algorithm For Constrained Optimization
Variable Metric Algorithm for Constrained Optimization (VMACO) is nonlinear computer program developed to calculate least value of function of n variables subject to general constraints, both equality and inequality. First set of constraints equality and remaining constraints inequalities. Program utilizes iterative method in seeking optimal solution. Written in ANSI Standard FORTRAN 77.
Frick, James D.↗
A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO
In this work, we provide a new approach for the efficient matrix-free application of the transpose of the Jacobian for the spectral element method for the adjoint-based solution of partial differential equation (PDE) constrained optimization. This results in optimizations of nonlinear PDEs using explicit integrators where the integration of the adjoint problem is not more expensive than the forward simulation. Solving PDE constrained optimization problems entails combining expertise from multiple areas, including simulation, computation of derivatives, and optimization. The Portable, Extensible Toolkit for Scientific computation (PETSc) together with its companion package, the Toolkit for Advanced Optimization (TAO), is an integrated numerical software library that contains an algorithmic/software stack for solving linear systems, nonlinear systems, ordinary differential equations, differential algebraic equations, and large-scale optimization problems and, as such, is an ideal tool for performing PDE-constrained optimization. This paper describes an efficient approach in which the software stack provided by PETSc/TAO can be used for large-scale nonlinear time-dependent problems. Time integration can involve a range of high-order methods, both implicit and explicit. The PDE-constrained optimization algorithm used is gradient-based and seamlessly integrated with the simulation of the physical problem.
97 MATHEMATICS AND COMPUTING↗
Uncertainty propagation and sensitivity analysis for constrained optimization of nuclear waste vitrification
Abstract The vitrification of high‐level waste (HLW) by heating a mixture of glass‐forming chemicals (GFCs) with the waste can be improved using a constrained optimization problem. This study explores how different uncertainty propagation (UP) methods implemented with the optimization process can affect the glass formulation of nuclear waste glasses. UP is the effort of propagating uncertain inputs through a system to understand and quantify output distributions. Uncertainty intervals are crafted from output distributions to inform the optimization algorithm. UP is often implemented with Monte Carlo (MC) sampling for large nonlinear systems, which can be difficult to implement within a constrained optimization algorithm that requires derivative information. Other UP methods often used for optimization under uncertainty (OUU) can be designed to work within an established constrained optimization framework. Methods of UP are evaluated in this study including iterative sampling approaches, first‐order approximations, and surrogate modeling with machine learning (ML). A method of dimensional reduction based on global sensitivity analysis is introduced to support the UP methods for the large dimensionality of the problem. Analytical UP methods able to achieve similar optimums 10 times faster than the baseline MC approach, and produce 93.9% similar output distributions are reported.
12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
arco (Assembled Resource-Constrained Optimization) [SWR-26-030]
Arco (Assembled Resource-Constrained Optimization) is a memory-smart optimization DSL and solver for LP and MIP problems on constrained hardware. The software is an optimization framework built around a KDL-based domain-specific language and a CLI compiler/solver. You write optimization models in .kdl files, and the arco CLI compiles, validates, inspects, and solves them. Language bindings (Python today, more planned) provide programmatic access to the same engine. Built for harder optimization problems on constrained resources, Arco is intentional about every allocation, careful with stack and heap behavior, and relentless about minimizing memory usage so more systems can run real workloads. Arco is built primarily for internal use within our organization. You are welcome to try it, but we make no guarantees about API stability or robustness at this stage
Sanchez Perez, Pedro Andres [National Laboratory o↗
Parallel Solver Framework for Mixed-Integer PDE-Constrained Optimization
ROL-PEBBL is a C++, MPI-based parallel code for mixed-integer PDE-constrained optimization (MIPDECO). In these problems we wish to optimize (control, design, etc.) physical systems, which must obey the laws of physics, when some of the decision variables must take integer values. ROL-PEBBL combines a code to efficiently search over integer choices (PEBBL = Parallel Enumeration Branch-and-Bound Library) and a code for efficient nonlinear optimization, including PDE-constrained optimization (ROL = Rapid Optimization Library). In this report, we summarize the design of ROL-PEBBL and initial applications/results. For an artificial source-inversion problem, finding sources of pollution on a grid from sparse samples, ROL-PEBBLs solution for the nest grid gave the best optimization guarantee for any general solver that gives both a solution and a quality guarantee.
97 MATHEMATICS AND COMPUTING↗
Globally Optimizing QAOA Circuit Depth for Constrained Optimization Problems
We develop a global variable substitution method that reduces n-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to 3-SAT and analyze the optimal quantum unitary circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark 3-SAT problems, we find that the upper bound of the unitary circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulation, which requires no decomposition.
3-SAT↗
Digital robust control law synthesis using constrained optimization
Development of digital robust control laws for active control of high performance flexible aircraft and large space structures is a research area of significant practical importance. The flexible system is typically modeled by a large order state space system of equations in order to accurately represent the dynamics. The active control law must satisy multiple conflicting design requirements and maintain certain stability margins, yet should be simple enough to be implementable on an onboard digital computer. Described here is an application of a generic digital control law synthesis procedure for such a system, using optimal control theory and constrained optimization technique. A linear quadratic Gaussian type cost function is minimized by updating the free parameters of the digital control law, while trying to satisfy a set of constraints on the design loads, responses and stability margins. Analytical expressions for the gradients of the cost function and the constraints with respect to the control law design variables are used to facilitate rapid numerical convergence. These gradients can be used for sensitivity study and may be integrated into a simultaneous structure and control optimization scheme.
Mukhopadhyay, Vivekananda↗
Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method
Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems cost-effectively, setting the stage for more advanced quantum simulations in chemistry.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Mixed-Strategy Chance Constrained Optimal Control
This paper presents a novel chance constrained optimal control (CCOC) algorithm that chooses a control action probabilistically. A CCOC problem is to find a control input that minimizes the expected cost while guaranteeing that the probability of violating a set of constraints is below a user-specified threshold. We show that a probabilistic control approach, which we refer to as a mixed control strategy, enables us to obtain a cost that is better than what deterministic control strategies can achieve when the CCOC problem is nonconvex. The resulting mixed-strategy CCOC problem turns out to be a convexification of the original nonconvex CCOC problem. Furthermore, we also show that a mixed control strategy only needs to "mix" up to two deterministic control actions in order to achieve optimality. Building upon an iterative dual optimization, the proposed algorithm quickly converges to the optimal mixed control strategy with a user-specified tolerance.
Ono, Masahiro↗
Compressing Vision Transformers in Geospatial Transfer Learning with Manifold-Constrained Optimization
Deploying geospatial foundation models on resource-constrained edge devices demands compact architectures that maintain high downstream performance. However, their large parameter counts and the accuracy loss often induced by compression limit practical adoption.In this work, we leverage manifold-constrained optimization framework DLRT to compress large vision transformer–based geospatial foundation models during transfer learning. By enforcing structured low-dimensional parameterizations aligned with downstream objectives, this approach achieves strong compression while preserving task-specific accuracy. We show that the method outperforms of-the-shelf low-rank methods as LoRA. Experiments on diverse geospatial benchmarks confirm substantial parameter reduction with minimal accuracy loss, enabling high-performing, on-device geospatial models.
Snyder, Thomas [Yale University]↗