Trust Region Subproblem Research Articles

Convergence of the block Lanczos method for the trust‐region subproblem in the hard case

SummaryThe trust‐region subproblem (TRS) plays a vital role in numerical optimization, numerical linear algebra, and many other applications. It is known that the TRS may have multiple optimal solutions in the hard case. In [Carmon and Duchi, SIAM Rev., 62 (2020), pp. 395–436], a block Lanczos method was proposed to solve the TRS in the hard case, and the convergence of the optimal objective value was established. However, the convergence of the KKT error as well as that of the approximate solution are still unknown for this method. In this paper, we give a more detailed convergence analysis on the block Lanczos method for the TRS in the hard case. First, we improve the convergence speed of the approximate objective value. Second, we derive the speed of the KKT error tends to zero. Third, we establish the convergence of the approximation solution, and show theoretically that the projected TRS obtained from the block Lanczos method will be close to the hard case more and more as the block Lanczos process proceeds. Numerical experiments illustrate the effectiveness of our theoretical results.

A Second-Order Finite-Difference Method for Derivative-Free Optimization

In this paper, a second-order finite-difference method is proposed for finding the second-order stationary point of derivative-free nonconvex unconstrained optimization problems. The forward-difference or the central-difference technique is used to approximate the gradient and Hessian matrix of objective function, respectively. The traditional trust-region framework is used, and we minimize the approximation trust region subproblem to obtain the search direction. The global convergence of the algorithm is given without the fully quadratic assumption. Numerical results show the effectiveness of the algorithm using the forward-difference and central-difference approximations.

Trust-Region Inverse Reinforcement Learning

This paper proposes a new unified inverse reinforcement learning (IRL) framework based on trust-region methods and a recently proposed Pontryagin differential programming (PDP) method in Jin et al. (2020), which aims to learn the parameters in both the system model and the cost function for three types of problems, namely, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -player nonzero-sum multistage games, 2-player zero-sum multistage games and 1-player optimal control, from demonstrated trajectories. Different from the existing frameworks using gradient to update learning parameters, our framework updates them with the candidate solution of trust-region subproblem ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {TRS}$</tex-math></inline-formula> ), where its required gradient and Hessian are obtained by differentiating Pontryagin's Maximum Principle (PMP) equations once and twice, respectively. The differentiated equations are shown to be equivalent to the PMP equations for affine-quadratic games / optimal control problems and can be solved by some explicit recursions. Extensive simulation examples and comparisons are presented to demonstrate the effectiveness of our proposed algorithm.

An adaptive simple model trust region algorithm based on new weak secant equations

&lt;abstract&gt;&lt;p&gt;In this work, we proposed a new trust region method for solving large-scale unconstrained optimization problems. The trust region subproblem with a simple form was constructed based on new weak secant equations, which utilized both gradient and function values and available information from the three most recent points. A modified Metropolis criterion was used to determine whether to accept the trial step, and an adaptive strategy was used to update the trust region radius. The global convergence and locally superlinearly convergence of the new algorithm were established under appropriate conditions. Numerical experiments showed that the proposed algorithm was effective.&lt;/p&gt;&lt;/abstract&gt;

Worst-Case Iteration Bounds for Log Barrier Methods on Problems with Nonconvex Constraints

Interior point methods (IPMs) that handle nonconvex constraints such as IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in the setting where the objective and constraints are thrice differentiable, and have Lipschitz first and second derivatives on the feasible region. We provide an IPM that, starting from a strictly feasible point, finds a μ-approximate Fritz John point by solving [Formula: see text] trust-region subproblems. For IPMs that handle nonlinear constraints, this result represents the first iteration bound with a polynomial dependence on [Formula: see text]. We also show how to use our method to find scaled-KKT points starting from an infeasible solution and improve on existing complexity bounds. Funding: This work was supported by Air Force Office of Scientific Research [9550-23-1-0242]. A significant portion of this work was done at Stanford where O. Hinder was supported by the PACCAR, Inc., Stanford Graduate Fellowship and the Dantzig-Lieberman fellowship.

Accelerating the Branch and Bound Algorithm for Solving the Extended Trust-Region Subproblems

Accelerating the Branch and Bound Algorithm for Solving the Extended Trust-Region Subproblems

Implementation of Asynchronous Distributed Gauss-Newton Optimization Algorithms for Uncertainty Quantification by Conditioning to Production Data

Summary Previous implementation of the distributed Gauss-Newton (DGN) optimization algorithm ran multiple optimization threads in parallel, employing a synchronous running mode (S-DGN). As a result, it waits for all simulations submitted in each iteration to complete, which may significantly degrade performance because a few simulations may run much longer than others, especially for time-consuming real-field cases. To overcome this limitation and thus improve the DGN optimizer’s execution, we propose two asynchronous DGN (A-DGN) optimization algorithms in this paper. The two A-DGN optimization algorithms are (1) the local-search algorithm (A-DGN-LS) to locate multiple maximum a-posteriori (MAP) estimates and (2) the integrated global-search algorithm with the randomized maximum likelihood (RML) method (A-DGN + RML) to generate hundreds of RML samples in parallel for uncertainty quantification. We propose using batch together with a checking time interval to control the optimization process. The A-DGN optimizers check the status of all running simulations after every checking time frame. The iteration index of each optimization thread is updated dynamically according to its simulation status. Thus, different optimization threads may have different iteration indices in the same batch. A new simulation case is proposed immediately once the simulation of an optimization thread is completed, without waiting for the completion of other simulations. We modified the training data set updating algorithm using each thread’s dynamically updated iteration index to implement the asynchronous running mode. We apply the modified QR decomposition method to estimate the sensitivity matrix at the best solution of each optimization thread by linear interpolation of all or a subset of the training data to avoid the issue of solving a linear system with a singular matrix because of insufficient training data points in early batches. A new simulation case (or search point) is generated by solving the Gauss-Newton (GN) trust-region subproblem (GNTRS) using the estimated sensitivity matrix. We developed a more efficient and robust GNTRS solver using eigenvalue decomposition (EVD). The proposed A-DGN optimization methods are tested and validated on a 2D analytical toy problem and a synthetic history-matching problem and then applied to a real-field deepwater reservoir model. Numerical tests confirm that the proposed A-DGN optimization methods can converge to solutions with matching quality comparable to those obtained by the S-DGN optimizers, saving on the time required for the optimizer to converge by a factor ranging from 1.3 to 2 when compared to the S-DGN optimizer depending on the problem. The new A-DGN optimization algorithms improve efficiency and robustness in solving history-matching or inversion problems, especially for uncertainty quantification of subsurface model parameters and production forecasts of real-field reservoirs by conditioning production data.

Semidefinite representable reformulations for two variants of the trust-region subproblem

Motivated by encouraging numerical results in Eltved and Burer (2023) [13], in this note we consider two specific variants of the trust-region subproblem and provide exact semidefinite representable reformulations. The first is over the intersection of two balls; the second is over the intersection of a ball and a special second-order conic representable set. Different from the technique developed in Eltved and Burer (2023) [13], the reformulations in this note are based on partitions of the feasible regions into sub-regions with known lifted convex hulls.

Positive semidefinite interval of matrix pencil and its applications to the generalized trust region subproblems

We are concerned with finding the set I⪰(A,B) of real values μ such that the matrix pencil A+μB is positive semidefinite. If A,B are not simultaneously diagonalizable via congruence (SDC), I⪰(A,B) either is empty or has only one value μ. When A,B are SDC, I⪰(A,B), if not empty, can be a singleton or an interval. Especially, if I⪰(A,B) is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval I≻(A,B). If A,B are both singular, then even I⪰(A,B) is an interval, its interior may not be I≻(A,B), but A,B are then decomposed as block diagonals of submatrices A1,B1 with B1 nonsingular such that I⪰(A,B)=I⪰(A1,B1). Applying I⪰(A,B), the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.

Performance Benchmarking of Different Methods to Solve Gauss-Newton Trust Region Subproblems

Summary The Gauss-Newton (GN) trust region optimization methods perform robustly but may introduce significant overhead cost when using the conventional matrix factorization method to solve the associated GN trust region subproblem (GNTRS). Solving a GNTRS involves solving a nonlinear equation using an iterative Newton-Raphson (NR) method. In each NR iteration, a symmetric linear system can be solved by different matrix factorization methods, including Cholesky decomposition (CD), eigenvalue decomposition (EVD), and singular value decomposition (SVD). Because CD fails to factorize a singular symmetric matrix, we propose solving a GNTRS using the robust EVD method. In this paper, we analyze the performances of different methods to solve a GNTRS using different matrix factorization subroutines in LAPACK with different options and settings. The cost of solving a GNTRS mainly depends on the number of observed data (m) and the number of uncertainty parameters (n). When n≤m, we recommend directly solving the original GNTRS with n variables. When n&amp;gt;m, we propose an indirect method that transforms the original GNTRS with n variables to a new problem with m unknowns. The proposed indirect method can significantly reduce the computational cost by dimension reduction. However, dimension reduction may introduce numerical errors, which, in turn, may result in accuracy degradation and cause failure of convergence using the popular iterative NR method. To further improve the overall performance, we introduce a numerical error indicator to terminate the iterative NR process when numerical errors become dominant. Finally, we benchmarked the performances of different approaches on a set of testing problems with different settings. Our results confirm that the GNTRS solver using the EVD method together with the modified NR method performs the best, being both robust (no failure for all testing problems) and efficient (consuming comparable CPU time to other methods).

Novel High-Precision and High-Robustness Localization Algorithm for Underwater-Environment-Monitoring Wireless Sensor Networks

In marine ecological environment monitoring, the acquisition of node location information is crucial, and the absence of location information can render the collected data meaningless. Compared to the rest of the distance-based localization methods, the received signal strength (RSS)-based localization technique has gained significant interest due to its low cost and the absence of time synchronization. However, the acoustic signal propagates in the complex and changeable aqueous medium, and, in addition to the time-varying path loss factor (PLF), there is often a certain absorption loss, which seriously deteriorates the localization accuracy of the RSS-based technique. To address the above challenges, we propose a novel high-precision and high-robustness localization (NHHL) algorithm that introduces an estimation parameter to conjointly estimate the marine node location and the ambient PLF. Firstly, the original non-convex localization problem is converted into an alternating nonnegative constrained least squares (ANCLS) framework with the unknown PLF and absorption loss, and a two-step localization method based on the primitive dual interior point method and block co-ordinate update method is presented to find the optimal solution. In the first step, the penalty function is utilized to reformulate the localization problem and find an approximate solution. Nevertheless, due to inherent errors, it is unable to approximate the constraint boundary and the global optimum solution. Subsequently, in the second step, the original localization problem is further transformed into a generalized trust region sub-problem (GTRS) framework, and the approximate solution of the interior point method is utilized as the initial estimation, and then iteratively solved by block co-ordinate update to obtain the precise location and PLF conjointly. Furthermore, the closed-form expression of the Cramér–Rao lower bound (CRLB) for the case of the unknown path loss factor and absorption loss is derived to evaluate the our NHHL algorithm. Finally, the simulation results demonstrate the superiority of the presented NHHL algorithm compared with the selected benchmark methods in various marine simulation scenarios.

Implicit Regularity and Linear Convergence Rates for the Generalized Trust-Region Subproblem

Implicit Regularity and Linear Convergence Rates for the Generalized Trust-Region Subproblem

Nonmonotone Alternative Direction Method Based on Simple Conic Model for Unconstrained Optimization

In this paper, we propose a nonmonotone Alternative Direction Method (ADM) based on simple conic model for unconstrained optimization. Unlike traditional trust region method, the subproblem in our method is a simple conic model, where the Hessian of the objective function is replaced by a scalar approximation, the trust region subproblem is solved by ADM which was first proposed by Zhu and Ni. When the trial point isn’t accepted by trust region, line search technique is used to find an acceptable point instead of resolving the trust region subproblem. The new method needs less memory capacitance and computational complexity. The global convergence of the algorithm is established under some mild conditions. Numerical results on a series of standard test problems are reported to show the new method is effective and robust.

Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource-constrained shortest-path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run-time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an [Formula: see text] algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the [Formula: see text] algorithm and a preprocessing procedure, which can be employed to accelerate the [Formula: see text] algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated [Formula: see text] algorithm and topological sorting outperform a general-purpose IP solver significantly. Moreover, the accelerated [Formula: see text] algorithm is able to outperform topological sorting for larger problem instances. We also give computational evidence that the performance of the superordinate trust-region algorithm may be improved if it is initialized with a solution obtained with the combinatorial integral approximation. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.1294 .

Local Optimality Conditions for a Family of Hidden Convex Optimization

Hidden convex optimization is a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we study a family of hidden convex optimization that joints the classical trust region subproblem (TRS) with convex optimization (CO). It also includes p-regularized subproblem (p &gt; 2) as a special case. We present a comprehensive study on local optimality conditions. In particular, a sufficient condition is given to ensure that there is at most one local nonglobal minimizer, and at this point, the standard second-order sufficient optimality condition is necessary. To our surprise, although (TRS) has at most one local nonglobal minimizer and (CO) has no local nonglobal minimizer, their joint problem could have any finite number of local nonglobal minimizers. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12171021, 12131004, and 11822103], the Beijing Natural Science Foundation [Grant Z180005], and the Fundamental Research Funds for the Central Universities. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2023.0089 .

Multistatic Localization by Differential Time Delays and Time Differences of Arrival in the Absence of Transmitter Position

We can only extract the differential time delay (DTD) measurements between the direct and reflected paths in multistatic localization when there is no synchronization in time between the transmitter and receiver and among the receivers. This paper first addresses the problem of multistatic localization of a fixed object when the transmitter position is not available by using the DTD measurements. We propose a two-step optimization method for jointly estimating the object and transmitter positions. In the first step, we formulate a non-convex constrained weighted least squares (CWLS) problem by transforming the DTD measurement model and introducing nuisance variables. Such a non-convex CWLS problem is then relaxed to a tractable convex semidefinite programming (SDP) problem by applying semidefinite relaxation. In the second step, the error coming from relaxation and approximation in the SDP solution is reduced iteratively through solving a generalized trust region subproblem (GTRS) in each iteration. If the receivers are synchronized such that the time difference of arrival (TDOA) measurements can be acquired in addition to DTD, we formulate a different CWLS problem by utilizing both DTD and TDOA measurements, which is solved by convex relaxation as well. The relaxed SDP problem can achieve the optimal solution of the CWLS problem, and further refinement is no longer needed. We conduct the mean square error (MSE) analysis to validate that both proposed methods are able to achieve the Cramer-Rao Lower Bound (CRLB) performance under small Gaussian noise, which is also validated by simulations.

PANTR: A Proximal Algorithm With Trust-Region Updates for Nonconvex Constrained Optimization

This work presents PANTR, an efficient solver for nonconvex constrained optimization problems, that is well-suited as an inner solver for an augmented Lagrangian method. The proposed scheme combines forward-backward iterations with solutions to trust-region subproblems: the former ensures global convergence, whereas the latter enables fast update directions. We discuss how the algorithm is able to exploit exact Hessian information of the smooth objective term through a linear Newton approximation, while benefiting from the structure of box-constraints or ℓ1-regularization. An open-source C++ implementation of PANTR is made available as part of the NLP solver library ALPAQA. Finally, the effectiveness of the proposed method is demonstrated in nonlinear model predictive control applications.

A trust-region based an active-set interior-point algorithm for fuzzy continuous Static Games

&lt;abstract&gt;&lt;p&gt;In this paper, a novel treatment for fuzzy continuous static games (FCSGs) is introduced. This treatment is based on the fact that, as well as having a fuzzy number, the fuzziness is applied to the control vectors to deal with high vagueness and imprecision in a continuous static game. The concept of the $ \alpha $-level set used for converting the FCSGs to a deterministic problem $ \alpha $-FCSGs. An active-set strategy is used with Newton's interior point method and a trust-region strategy to insure global convergence for deterministic $ \alpha $-FCSGs problems from any starting point. A reduced Hessian technique is used to overcome the difficulty of having an infeasible trust-region subproblem. The active-set interior-point trust-region algorithm has new features; it is easy to implement and has rapid convergence. Preliminary numerical results are reported.&lt;/p&gt;&lt;/abstract&gt;

Non-quadratic extension of homogeneous S-lemma and its applications in optimization

In this paper, we propose a non-quadratic extension of homogeneous S-lemma under some conditions. Then we apply the extended homogeneous S-lemma to reveal the hidden convexity property of the generalized Kantorovich inequality problem and the homogeneously lifted two-sided extended trust-region subproblem.

Algorithm 1030: SC-SR1: MATLAB Software for Limited-memory SR1 Trust-region Methods

We present a MATLAB implementation of the symmetric rank-one (SC-SR1) method that solves trust-region subproblems when a limited-memory symmetric rank-one (L-SR1) matrix is used in place of the true Hessian matrix, which can be used for large-scale optimization. The method takes advantage of two shape-changing norms [Burdakov and Yuan 2002 ; Burdakov et al. 2017 ] to decompose the trust-region subproblem into two separate problems. Using one of the proposed norms, the resulting subproblems have closed-form solutions. Meanwhile, using the other proposed norm, one of the resulting subproblems has a closed-form solution while the other is easily solvable using techniques that exploit the structure of L-SR1 matrices. Numerical results suggest that the SC-SR1 method is able to solve trust-region subproblems to high accuracy even in the so-called “hard case.” When integrated into a trust-region algorithm, extensive numerical experiments suggest that the proposed algorithms perform well, when compared with widely used solvers, such as truncated conjugate-gradients.