Robust Nonlinear Optimization Research Articles

A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle

A Primal-Worst-Equals-Dual-Best Perspective on Robust and Distributionally Robust Optimization In the paper “A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle,” Jianzhe Zhen, Daniel Kuhn, and Wolfram Wiesemann develop a generalized “primal-worst-equals-dual-best” principle that establishes strong duality between semi-infinite primal worst and nonconvex dual best formulations of robust and distributionally robust nonlinear optimization problems. Their theory offers an alternative characterization of (distributionally) robust optimization problems that bypasses the need to mobilize the machinery of abstract semi-infinite duality theory. The paper will be of interest to researchers and practitioners in the field of optimization under uncertainty.

Approximating Linear Programming by Geometric Programming and Its Application to Urban Planning

In this study, we approximated linear programming by geometric programming; the developed method converts linear programming to geometric programming. We applied the developed method to Neighbourhood planning, a vital aspect of urban planning, and obtained the optimal cost of Neighbourhood designs. The method demonstrated that geometric programming is a robust non-linear optimization model that can be extended to approximate linear optimization problems. This method has obvious advantages in the sense that it allows every decision variable to contribute to the optimal objective function. This is not the case with the known regular Simplex method and the Interior Point Algorithm of solution to linear programming which assign zeros to some variables when the matrix of the non-basic variables is rectangular or when some of the non-basic variables did not enter the basis. The developed method was used to find the global optimal solution, optimal primal and dual decision variables. The solution was better compared to the linear programming method via Simplex method or Interior Point Algorithm because it achieved the global optimal solution. We observed that in addition to achieving the global optimal solution, we obtained the optimal dual decision variables which was absent in the other methods and all the primal decision variables have value against the other methods that assigned some of the variables with zeroes.

Duality Results for a Class of Constrained Robust Nonlinear Optimization Problems

In this paper, we establish various results of duality for a new class of constrained robust nonlinear optimization problems. For this new class of problems, involving functionals of (path-independent) curvilinear integral type and mixed constraints governed by partial derivatives of second order and uncertain data, we formulate and study Wolfe, Mond-Weir and mixed type robust dual optimization problems. In this regard, by considering the concept of convex curvilinear integral vector functional, determined by controlled second-order Lagrangians including uncertain data, and the notion of robust weak efficient solution associated with the considered problem, we create a new mathematical context to state and prove the duality theorems. Furthermore, an illustrative application is presented.

Outer Approximation for Mixed-Integer Nonlinear Robust Optimization

Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integer nonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem.In our approach, we model a robust optimization problem as a nonsmooth mixed-integer nonlinear problem and tackle it by an outer approximation method that requires only inexact function values and subgradients. To deal with the arising nonlinear subproblems, we render an adaptive bundle method applicable to this setting and extend it to generate cutting planes, which are valid up to a known precision. Relying on its convergence to approximate critical points, we prove, as a consequence, finite convergence of the outer approximation algorithm.As an application, we study the gas transport problem under uncertainties in demand and physical parameters on realistic instances and provide computational results demonstrating the efficiency of our method.

Technical Note—Dual Approach for Two-Stage Robust Nonlinear Optimization

In “Dual Approach for Two-Stage Robust Nonlinear Optimization,” de Ruiter, Zhen, and den Hertog study adjustable robust minimization problems where the objective or constraints depend in a convex way on the adjustable variables. They reformulate the original adjustable robust nonlinear problem with a polyhedral uncertainty set into an equivalent adjustable robust linear problem, for which all existing approaches for adjustable robust linear problems can be used. The reformulation is obtained by first dualizing over the adjustable variables and then over the uncertain parameters. The polyhedral structure of the uncertainty set then appears in the linear constraints of the dualized problem, and the nonlinear functions of the adjustable variables in the original problem appear in the uncertainty set of the dualized problem. The authors show how to recover linear decision rules to the original primal problem and how to generate bounds on its optimal objective value.

Adaptive Bundle Methods for Nonlinear Robust Optimization

Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the approaches that do exist impose restrictive assumptions on the problem structure. We present an adaptive bundle method for nonlinear and nonconvex robust optimization problems with a suitable notion of inexactness in function values and subgradients. As the worst-case evaluation requires a global solution to the adversarial problem, it is a main challenge in a general nonconvex nonlinear setting. Moreover, computing elements of an ε-perturbation of the Clarke subdifferential in the [Formula: see text]-norm sense is in general prohibitive for this class of problems. In this article, instead of developing an entirely new bundle concept, we demonstrate how existing approaches, such as Noll’s bundle method for nonconvex minimization with inexact information [Noll D (2013) Bundle method for non-convex minimization with inexact subgradients and function values. Computational and Analytical Mathematics, Springer Proceedings Mathematics, vol. 50 (Springer, New York), 555–592.] can be modified to be able to cope with this situation. Extending the nonconvex bundle concept to the case of robust optimization in this way, we prove convergence under two assumptions: first, that the objective function is lower C1 and, second, that approximately optimal solutions to the adversarial maximization problem are available. The proposed method is, hence, applicable to a rather general setting of nonlinear robust optimization problems. In particular, we do not rely on a specific structure of the adversary’s constraints. The considered class of robust optimization problems covers the case that the worst-case adversary only needs to be evaluated up to a certain precision. One possibility to evaluate the worst case with the desired degree of precision is the use of techniques from mixed-integer linear programming. We investigate the procedure on some analytic examples. As applications, we study the gas transport problem under uncertainties in demand and in physical parameters that affect pressure losses in the pipes. Computational results for examples in large realistic gas network instances demonstrate the applicability as well as the efficiency of the method. Summary of Contribution: Nonlinear robust optimization is a relevant field of research as real-world optimization problems usually suffer from not precisely known parameters, for example, physical parameters that cannot be measured exactly. Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the methods that do exist impose restrictive assumptions on the problem structure. Writing nonlinear robust optimization tasks in minimax form, in principle, bundle methods can be used to solve the resulting nonsmooth problem. However, there are a number of difficulties to overcome. First, the inner adversarial problem needs to be solved to global optimality, which is a major challenge in a general nonconvex nonlinear setting. In order to cope with this, an adaptive solution approach, which allows for inexactness, is required. A second challenge is then that the computation of elements from an ε-neighborhood of the Clarke subdifferential is, in general, prohibitive. We show how an existing bundle concept by D. Noll for nonconvex problems with inexactness in function values and subgradients can be adapted to this situation. The resulting method only requires availability of approximate worst-case evaluations, and in particular, it does not rely on a specific structure of the adversarial constraints. To evaluate the worst case with the desired degree of precision, one possibility is the use of techniques from mixed-integer linear programming. In the course of the paper, we discuss convergence properties of the resulting method and demonstrate its efficiency by means of robust gas transport problems.

An interval sequential linear programming for nonlinear robust optimization problems

In this paper, interval sequential linear programming (ISLP) is proposed to solve nonlinear robust optimization (RO). The main idea of the programming is to transform the uncertain optimization into several possibility-sensitivity analyses and deterministic linear optimization problems that are sequentially solved. At each cycle, a possibility-sensitivity analysis method is proposed to obtain the approximate partial derivatives of the uncertain constraints at the current design point, based on which a deterministic linear optimization model is constructed and the design point is updated by solving the linear optimization. Moreover, an iterative mechanism is created to adaptively update the design space and improve the convergence rate. Finally, two numerical examples and two practical engineering problems are applied to verify the accuracy and efficiency of the proposed method.

A general data-driven nonlinear robust optimization framework based on statistic limit and principal component analysis

A general robust optimization framework is proposed to solve nonlinear programming under uncertainty. A compact convex data-driven uncertainty set is first conducted by leveraging the combined index statistic limit technique. It can effectively capture the correlations among uncertain variables by using the principal component analysis model, and eliminate the noise in massive uncertainty data. Based on the proposed uncertainty set, linearization is taken to approximate nonlinear optimization with inequality-only constraints by using first-order Taylor approximation. By using the implicit function theorem, it is extended to a general formulation involving both inequality and equality constraints. Due to the potential limitation of first-order Taylor approximation, an iterative algorithm is designed to realize multiple linearization to search for a global robust solution under large perturbation. The efficiency of the proposed approach is verified on simulated numerical experiences, and the proposed method is applied to the industrial process of gold cyanidation leaching.

A generalized cutting‐set approach for nonlinear robust optimization in process systems engineering

AbstractWe propose a novel computational framework for the robust optimization of highly nonlinear, non‐convex models that possess uncertainty in their parameter data. The proposed method is a generalization of the robust cutting‐set algorithm that can handle models containing irremovable equality constraints, as is often the case with models in the process systems engineering domain. Additionally, we accommodate general forms of decision rules to facilitate recourse in second‐stage (control) variables. In particular, we compare and contrast the use of various types of decision rules, including quadratic ones, which we show in certain examples to be able to decrease the overall price of robustness. Our proposed approach is demonstrated on three process flow sheet models, including a relatively complex model for amine‐based CO2 capture. We thus verify that the generalization of the robust cutting‐set algorithm allows for the facile identification of robust feasible designs for process systems of practical relevance.

Robust Autocalibration of Triaxial Magnetometers

Self-calibration of a magnetometer usually requires controlled magnetic environment as the calibration output can be affected by field distortions from nearby magnetic objects. In this article, we develop a two-stage method that can accurately self-calibrate magnetometer from measurements containing anomalous readings due to local magnetic disturbances. The method proceeds by robustly fitting an ellipsoid to measurement data via L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm convex optimization, yielding initial model variables that are less prone to magnetic disruptions. These are then served as a starting point for robust nonlinear least-squares optimization, which refines the magnetometer model to minimize sensor estimation errors while suppressing heavy anomalies. Synthetic and real experimental results are provided to demonstrate improved accuracy of the proposed method in the presence of outliers. We additionally show empirically that the method is directly applicable to self-calibration of three-axis accelerometers.

Robust treatment planning with 4D intensity modulated carbon ion therapy for multiple targets in stage IV non-small cell lung cancer

Intensity modulated particle therapy (IMPT) with carbon ions can generate highly conformal treatment plans; however, IMPT is limited in robustness against range and positioning uncertainty. This is particularly true for moving targets, even though all motion states of a 4DCT are considered in 4D-IMPT. Here, we expand 4D-IMPT to include robust non-linear RBE-weighted optimization to explore its potential in improving plan robustness and sparing critical organs. In this study, robust 4D-optimization—based on worst-case optimization on 9 scenarios—was compared to conventional 4D-optimization with PTV margins using 4D dose calculation and robustness analysis for 21 uncertainty scenarios. Slice-by-slice rescanning was used for motion mitigation. Both 4D-optimization strategies were tested on a cohort of 8 multi-lesion lung cancer patients with the goal of prioritizing OAR sparing in a hypofractionated treatment plan. Planning objectives were to keep the OAR volume doses below corresponding limits while simultaneously achieve CTV coverage with D95% ≥ 95 %. For the conventional plans, average D95% was at 98.7% which fulfilled the target objective in 83.2% of scenarios. For the robust plans, average D95% was reduced to 97.6% which still fulfilled the target objective in 80.7% of cases, but led to significantly improved overall OAR sparing: Volume doses were below the limits in 96.2% of cases for the conventional and 99.5% for the robust plans. When considering the particularly critical smaller airways only, fulfillment rates could be increased from 76.2% to 96% for the robust plans. This study has shown that plan robustness of 4D-IMPT could be improved by using robust 4D-optimization, offering greater control over uncertainties in the actual delivered dose. In some cases, this required sacrificing target coverage for the benefit of better OAR sparing.

An Approximation Scheme for Distributionally Robust Nonlinear Optimization

We consider distributionally robust optimization problems (DROPs) with nonlinear and nonconcave dependence on uncertain parameters. The DROP can be written as a nonsmooth, nonlinear program with a ...

Two-Stage Robust Optimization of Water Treatment Network Design and Operations under Uncertainty

This paper presents a study of two-stage adaptive nonlinear robust optimization in dealing with uncertainty in the optimal design and operation of water treatment systems. This work aims to obtain ...

The application of artificial neural networks in metabolomics: a historical perspective.

Metabolomics data, with its complex covariance structure, is typically modelled by projection-based machine learning (ML) methods such as partial least squares (PLS) regression, which project data into a latent structure. Biological data are often non-linear, so it is reasonable to hypothesize that metabolomics data may also have a non-linear latent structure, which in turn would be best modelled using non-linear equations. A non-linear ML method with a similar projection equation structure to PLS is artificial neural networks (ANNs). While ANNs were first applied to metabolic profiling data in the 1990s, the lack of community acceptance combined with limitations in computational capacity and the lack of volume of data for robust non-linear model optimisation inhibited their widespread use. Due to recent advances in computational power, modelling improvements, community acceptance, and the more demanding needs for data science, ANNs have made a recent resurgence in interest across research communities, including a small yet growing usage in metabolomics. As metabolomics experiments become more complex and start to be integrated with other omics data, there is potential for ANNs to become a viable alternative to linear projection methods. We aim to first describe ANNs and their structural equivalence to linear projection-based methods, including PLS regression. We then review the historical, current, and future uses of ANNs in the field of metabolomics. Is metabolomics ready for the return of artificial neural networks?

PO-0910: Nonlinear robust optimization methods for 4D treatment planning in carbon ion therapy

PO-0910: Nonlinear robust optimization methods for 4D treatment planning in carbon ion therapy

Prediction Equation of Spectral Acceleration Responses in Low-to-Moderate Seismic Regions using Domestic and Overseas Earthquake Records

This study develops an empirical prediction equation of spectral acceleration responses of earthquakes which can induce structural damages. Ground motion records representing hazards of low-to-moderate seismic regions were selected and organized with several influential factors affecting the response spectra. The empirical equation and estimator coefficients for acceleration response spectra were then proposed using a robust nonlinear optimization coupled with a regression analysis. For analytical verification of the prediction equation, response spectra used for low-to-moderate seismic regions were estimated and the predicted results were comparatively evaluated with measured response spectra. As a result, the predicted shapes of response spectra can simulate the graphical shapes of measured data with high accuracy and most of predicted results are distributed inside range of correlation of variation (COV) of 30% from perfectly correlated lines.

Sterilization network design

With the increasing costs of health care, service providers are looking for innovative ways to cut cost without sacrificing high service quality. Resource pooling is a promising trend in different health care areas, and sterilization is no exception. Specialized medical sterilization centers (SCs) that offer reusable medical device (RMD) sterilization services for hospitals have the potential to cut cost and improve efficiency through better utilization of resources, risk-pooling and economies of scale. However, it is unclear whether the resulting cost savings can offset the additional transportation costs and the operational complications associated with a centralized system. We compare three schemes to organize RMD sterilization in a group of hospitals: fully distributed, centralized processing, and centralized processing and stock keeping. The sterilization network design problem is formulated as a mixed-integer concave minimization program (MISOCP) that considers economies of scale, service level requirements and variable demand, with the objective of minimizing capacity, transportation, and inventory holding costs. The mathematical model is reformulated as a mixed-integer second-order cone program with a piecewise-linear cost function so it can be solved efficiently. We also consider the case when the first two moments of the RMD demand distributions are not known with certainty but are rather estimated based on sample data. We show that, with proper selection of the uncertainty structure, the robust nonlinear optimization problem can be tractably reformulated as a MISOCP as well. Testing on a realistic case study under different scenarios reveals that significant cost savings can be achieved by consolidating sterilization services. Compared to the distributed scheme, we found that cost saving in the second scheme is attributed primarily to improved resource utilization and economies of scale. Risk-pooling in the last scheme results in an additional small cost saving that has to be weighed against other operational and legal considerations.

Nonlinear robust optimization for process design

A novel robust optimization framework is proposed to address general nonlinear problems in process design. Local linearization is taken with respect to the uncertain parameters around multiple realizations of the uncertainty, and an iterative algorithm is implemented to solve the problem. Furthermore, the proposed methodology can handle different categories of problems according to the complexity of the problems. First, inequality‐only constrained optimization problem as studied in most existing robust optimization methods can be addressed. Second, the proposed framework can deal with problems with equality constraint associated with uncertain parameters. In the final case, we investigate problems with operation variables which can be adjusted according to the realizations of uncertainty. A local affinely adjustable decision rule is adopted for the operation variables (i.e., an affine function of the uncertain parameter). Different applications corresponding to different classes of problems are used to demonstrate the effectiveness of the proposed nonlinear robust optimization framework. © 2017 American Institute of Chemical Engineers AIChE J, 64: 481–494, 2018

Model Order Reduction Techniques with a Posteriori Error Control for Nonlinear Robust Optimization Governed by Partial Differential Equations

We consider a nonlinear optimization problem governed by partial differential equations with uncertain parameters. It is addressed by a robust worst case formulation. The resulting optimization problem is of bilevel structure and is difficult to treat numerically. We propose an approximate robust formulation that employs linear and quadratic approximations. To speed up the computation, reduced order models based on proper orthogonal decomposition in combination with a posteriori error estimators are developed. The proposed strategy is then applied to the optimal placement of a permanent magnet in the rotor of a synchronous machine with moving rotor. Numerical results are presented to validate the presented approach.

Robust optimization using hybrid differential evolution and sequential quadratic programming

Although robust optimization can find solutions to engineering problems with uncertainty, the computational inefficiency of robust optimization methods is still a major concern. In this article, a new hybrid robust optimization method, namely differential evolution–sequential quadratic programming–robust optimization (DE-SQP-RO), is presented to find global robust optima for nonlinear robust optimization problems. The proposed algorithm is conducted under the structure of DE-RO, with SQP-RO acting as a local optimizer. Two criteria and switch indices are developed to indicate when the algorithm should switch from DE-RO to SQP-RO and vice versa. One numerical and two engineering examples are tested to demonstrate the applicability of the proposed algorithm. The results show that the hybrid algorithm uses 29–45% of the number of function evaluations required by DE-RO, and the robust solutions obtained by the hybrid algorithm are even better for certain examples.