Derivatives Of Objective Function Research Articles

Higher-order optimality conditions with separated derivatives and sensitivity analysis for set-valued optimization

In this paper, we establish optimality conditions and sensitivity analysis of set-valued optimization problems in terms of higher-order radial derivatives. First, we obtain the optimality conditions with separated derivatives for a set-valued optimization problem, here separated derivatives means the derivatives of objective and constraint functions are different. Then, some duality theorems for a mixed type of primal-dual set-valued optimization problem are gained. Finally, several results concerning higher-order sensitivity analysis are presented. The main results of this paper are illustrated by some concrete examples.

Higher-Order Radial Hadamard Directional Derivatives and Applications to Set-Valued Equilibrium Problems

In this paper, we first introduce the higher-order lower and upper radial Hadamard directional derivatives for set-valued maps, which can capture the global information of concerning maps, and discuss the relationships with other existed higher-order derivatives. Second, based on the introduced derivatives, we establish optimality conditions for Henig efficient solutions of a set-valued equilibrium problem with constraints. Particularly, the optimality conditions hold in the case where the derivatives of objective and constraint functions are separated. Finally, we give some duality theorems for a mixed type of primal-dual set-valued equilibrium problem. The main results of this paper are illustrated by several concrete examples.

INTUITIONISTIC ROBUST CLUSTERING FOR SEGMENTATION OF LESIONS IN DERMATOSCOPIC IMAGES

This paper presents the formulation of the intuitive fuzzy clustering algorithm to be robust to atypical data present in dermoscopic images and to delimit the affected area. This algorithm is formulated from the objective function derivation for memberships update, to integrate an m-redescending estimator influence function. Experimentation shows an accuracy of 95% with the proposal algorithm with respect to other clustering algorithms to perform delimitations, in addition the iterations number is considerably reduced. Keywords: Robust Intuitionistic Fuzzy Clustering, Dermoscopic Images, Delimitations of Lesions, M-redescending Estimator

Novel lightweight connecting bracket design with multiple performance constraints based on optimization and verification process

<abstract> <p>In this paper, a complete optimization design verification process is proposed and a novel structure of connecting brackets is presented, solving the fatigue failure of chassis connecting brackets operating on harsh roads. First, an endurance road test and fatigue life analysis were applied to the truck equipped with the original brackets, verifying the fatigue damage of the structure. Based on the solid isotropic material with penalization method, a novel lightweight connecting bracket layout was obtained by using the method of moving asymptotes (MMA) for topology optimization under multiple working conditions with multiple performance constraints. Moreover, the derivatives of objective and constraint functions concerning design variables were applied for the MMA. Considering manufacturability and functionality, the improved model based on the topology optimization results was further optimized by size optimization. Finally, fatigue life analysis and an endurance road test were conducted using the optimal design. Compared with the original structure, the novel brackets showed better stiffness, strength and fatigue performance while reducing the total mass by 15.2%. The whole optimization and validation process can provide practical ideas and value for developing multi-performance suspensions in the pre-product development stage.</p> </abstract>

TinyAD: Automatic Differentiation in Geometry Processing Made Simple

AbstractNon‐linear optimization is essential to many areas of geometry processing research. However, when experimenting with different problem formulations or when prototyping new algorithms, a major practical obstacle is the need to figure out derivatives of objective functions, especially when second‐order derivatives are required. Deriving and manually implementing gradients and Hessians is both time‐consuming and error‐prone. Automatic differentiation techniques address this problem, but can introduce a diverse set of obstacles themselves, e.g. limiting the set of supported language features, imposing restrictions on a program's control flow, incurring a significant run time overhead, or making it hard to exploit sparsity patterns common in geometry processing. We show that for many geometric problems, in particular on meshes, the simplest form of forward‐mode automatic differentiation is not only the most flexible, but also actually the most efficient choice. We introduce TinyAD: a lightweight C++ library that automatically computes gradients and Hessians, in particular of sparse problems, by differentiating small (tiny) sub‐problems. Its simplicity enables easy integration; no restrictions on, e.g., looping and branching are imposed. TinyAD provides the basic ingredients to quickly implement first and second order Newton‐style solvers, allowing for flexible adjustment of both problem formulations and solver details. By showcasing compact implementations of methods from parametrization, deformation, and direction field design, we demonstrate how TinyAD lowers the barrier to exploring non‐linear optimization techniques. This enables not only fast prototyping of new research ideas, but also improves replicability of existing algorithms in geometry processing. TinyAD is available to the community as an open source library.

Robust topology optimization methodology for continuum structures under probabilistic and fuzzy uncertainties

AbstractOwing to the variations in geometric dimensions, material properties and external loads in engineering applications, robust topology optimization (RTO) has garnered increasing attention in recent years to account for the uncertain behaviors during the preliminary concept design phases. This paper presents a hybrid RTO method to simultaneously resolve the epistemic and aleatory uncertainties. First, based on the probabilistic and fuzzy methodologies, the hybrid RTO model is formulated with nested double optimization loops using Monte Carlo simulations. Second, an efficient iterative method is proposed based on the perturbation method to accelerate the rate of convergence of the proposed hybrid RTO model. The derivatives of the hybrid robust compliance objective function with respect to the deterministic design variables, random parameters, and fuzzy parameters are then derived using the adjoint variable method. Finally, a T‐shaped beam design, an L‐shaped beam design, and a three‐dimensional cantilever beam design are tested to validate the proposed hybrid RTO method.

Second-Order characterizations for set-valued equilibrium problems with variable ordering structures

<p style='text-indent:20px;'>The concepts of weakly efficient solutions and globally efficient solutions are introduced for constrained set-valued equilibrium problems with variable ordering structures. By applying the second-order tangent epiderivative and a nonlinear functional, necessary optimality conditions for weakly efficient solutions and globally efficient solutions are established without any convexity assumption. Under the cone-convexity of the objective and constraint functions, sufficient optimality conditions are given. In addition, the tangent derivatives of objective and constraint functions are separated. Simultaneously, a unified necessary and sufficient optimality conditions for weakly efficient solutions is derived, and the same goes for globally efficient solutions. In particular, we give specific examples to illustrate the optimality conditions, respectively.</p>

A Modified Multiple Shooting Algorithm for Parameter Estimation in ODEs Using Adjoint Sensitivity Analysis

To increase the predictive power of a model, one needs to estimate its unknown parameters. Almost all parameter estimation techniques in ordinary differential equation models suffer from either a small convergence region or enormous computational cost. The method of multiple shooting, on the other hand, takes its place in between these two extremes. The computational cost of the algorithm is mostly due to the calculation of directional derivatives of objective and constraint functions. Here we modify the multiple shooting algorithm to use the adjoint method in calculating these derivatives. In the literature, this method is known to be a more stable and computationally efficient way of computing gradients of scalar functions. A predator-prey system is used to show the performance of the method and supply all necessary information for a successful and efficient implementation.

New higher-order weak lower inner epiderivatives and application to Karush–Kuhn–Tucker necessary optimality conditions in set-valued optimization

The purpose of the paper is to establish higher-order Karush–Kuhn–Tucker higher-order necessary optimality conditions for set-valued optimization where the derivatives of objective and constraint functions are separated. We first introduce concepts of higher-order weak lower inner epiderivatives for set-valued maps and discuss some useful properties about new epiderivatives, for instance, convexity, subadditivity and chain rule. With the help of the new concept and its properties, we establish higher-order Karush–Kuhn–Tucker necessary optimality conditions which is the classical type Karush–Kuhn–Tucker optimality conditions and improve and enhance some recent existing results in the literatures. Several examples are provided to illustrate our results. Finally, we provide weak and strong duality theorems in set-valued optimization.

Monte-Carlo and sensitivity transport models for domain deformation

We address the question of evaluating shape derivatives of objective functions for radiative-transfer engineering involving semi-transparent media. After recalling the standard Monte-Carlo approach to sensitivity estimation and its current limitations, a new method is presented for the specific case of geometrical sensitivities. This method is then tested on configurations with multiple-scattering and absorbing (non-emitting) semi-transparent medium. A new geometrical sensitivity algorithm is presented with full details in order to extend, on several examples, its implementation in complex geometries.

A derivative-free algorithm for spherically constrained optimization

Spherically constrained optimization, which minimizes an objective function on a unit sphere, has wide applications in numerical multilinear algebra, signal processing, solid mechanics, etc. In this paper, we consider a certain case that the derivatives of the objective function are unavailable. This case arises frequently in computational science, chemistry, physics, and other enormous areas. To explore the spherical structure of the above problem, we apply the Cayley transform to preserve iterates on the sphere and propose a derivative-free algorithm, which employs a simple model-based trust-region framework. Under mild conditions, global convergence of the proposed algorithm is established. Preliminary numerical experiments illustrate the promising performances of our algorithm.

Two-order approximate and large stepsize numerical direction based on the quadratic hypothesis and fitting method

Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems ʼ objective functions have no derivatives. According to former research studies, some search directions are obtained using the quadratic hypothesis of objective functions. Based on derivatives, quadratic function assumptions, and directional derivatives, the computational formulas of numerical first-order partial derivatives, second-order partial derivatives, and numerical second-order mixed partial derivatives were constructed. Based on the coordinate transformation relation, a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction. A numerical algorithm was proposed, taking the second order approximation direction as an example. A large stepsize numerical algorithm based on coordinate transformation was proposed. Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function. The numerical second order approximation direction with the numerical mixed partial derivatives showed good results. Its calculated amount is 0.2843 % of that of without second-order mixed partial derivative. In the process of rotating the local coordinate system 360 ° , because the objective function is more complex than the quadratic function, if the numerical direction derivative is used instead of the analytic partial derivative, the optimization direction varies with a range of 103.05 ° . Because theoretical error is in the numerical negative gradient direction, the calculation with the coordinate transformation is 94.71 % less than the calculation without coordinate transformation. If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation, the sawtooth phenomenon occurs. When each numerical mixed partial derivative takes more than one point, the optimization results cannot be improved. The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained, but does not require derivability and does not take into account truncation error and rounding error. Thus, the application scopes of many optimization methods are extended.

Downhole Microseismic Source Location Based on a Multi‐Dimensional DIRECT Algorithm for Unconventional Oil and Gas Reservoir Exploration

Downhole microseismic data has the significant advantages of high signal‐to‐noise ratio and well‐developed P and S waves and the core component of microseismic monitoring is microseismic event location associated with hydraulic fracturing in a relatively high confidence level and accuracy. In this study, we present a multidimensional DIRECT inversion method for microseismic locations and applicability tests over modeling data based on a downhole microseismic monitoring system. Synthetic tests inidcate that the objective function of locations can be defined as a multi‐dimensional matrix space by employing the global optimization DIRECT algorithm, because it can be run without the initial value and objective function derivation, and the discretely scattered objective points lead to an expeditious contraction of objective functions in each dimension. This study shows that the DIRECT algorithm can be extensively applied in real downhole microseismic monitoring data from hydraulic fracturing completions. Therefore, the methodology, based on a multidimensional DIRECT algorithm, can provide significant high accuracy and convergent efficiency as well as robust computation for interpretable spatiotemporal microseismic evolution, which is more suitable for real‐time processing of a large amount of downhole microseismic monitoring data.

Shape optimization of a coupled thermal fluid–structure problem in a level set mesh evolution framework

Hadamard’s method of shape differentiation is applied to topology optimization of a weakly coupled three physics problem. The coupling is weak because the equations involved are solved consecutively, namely the steady state Navier–Stokes equations for the fluid domain, first, the convection diffusion equation for the whole domain, second, and the linear thermo-elasticity system in the solid domain, third. Shape sensitivities are derived in a fully Lagrangian setting which allows us to obtain shape derivatives of general objective functions. An emphasis is given on the derivation of the adjoint interface condition dual to the one of equality of the normal stresses at the fluid solid interface. The arguments allowing to obtain this surprising condition are specifically detailed on a simplified scalar problem. Numerical test cases are presented using the level set mesh evolution framework of Allaire et al. (Appl Mech Eng 282:22–53, 2014). It is demonstrated how the implementation enables to treat a variety of shape optimization problems.

Adaptive Regularization Algorithms with Inexact Evaluations for Nonconvex Optimization

A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is $\beta$-H\"{o}lder continuous. It features a very flexible adaptive mechanism for determining the inexactness which is allowed, at each iteration, when computing objective function values and derivatives. The complexity analysis covers arbitrary optimality order and arbitrary degree of available approximate derivatives. It extends results of Cartis, Gould and Toint (2018) on the evaluation complexity to the inexact case: if a $q$th order minimizer is sought using approximations to the first $p$ derivatives, it is proved that a suitable approximate minimizer within $\epsilon$ is computed by the proposed algorithm in at most $O(\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ iterations and at most $O(|\log(\epsilon)|\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ approximate evaluations. An algorithmic variant, although more rigid in practice, can be proved to find such an approximate minimizer in $O(|\log(\epsilon)|+\epsilon^{-\frac{p+\beta}{p-q+\beta}})$ evaluations.While the proposed framework remains so far conceptual for high degrees and orders, it is shown to yield simple and computationally realistic inexact methods when specialized to the unconstrained and bound-constrained first- and second-order cases. The deterministic complexity results are finally extended to the stochastic context, yielding adaptive sample-size rules for subsampling methods typical of machine learning.

Derivative-free robust optimization by outer approximations

We develop an algorithm for minimax problems that arise in robust optimization in the absence of objective function derivatives. The algorithm utilizes an extension of methods for inexact outer approximation in sampling a potentially infinite-cardinality uncertainty set. Clarke stationarity of the algorithm output is established alongside desirable features of the model-based trust-region subproblems encountered. We demonstrate the practical benefits of the algorithm on a new class of test problems.

Transient Optimization of Natural Gas Networks Using Intelligent Algorithms

Compressor stations in natural gas networks should perform such that time-varying demands of customers are fulfilled while all of the system constraints are satisfied. Power consumption of compressor stations impose the most operational cost to a gas network so their optimal performance will lead to significant money saving. In this paper, the gas network transient optimization problem is addressed. The objective function is the sum of the compressor's power consumption that should be minimized where compressor speeds and the value status are decision variables. This objective function is nonlinear which is subjected to nonlinear and combinatorial constraints including both discrete and continuous variables. To handle this challenging optimization problem, a novel approach based on using two different structure intelligent algorithms, namely the particle swarm optimization (PSO) and cultural algorithm (CA), is utilized to find the optimum of the decision variables. This approach removes the necessity of finding an explicit expression for the power consumption of compressors as a function of decision variables as well as the calculation of objective function derivatives. The objective function and constraints are evaluated in the transient condition by a fully implicit finite difference numerical method. The proposed approach is applied on a real gas network where simulation results confirm its accuracy and efficiency.

Employing the Time-Domain Unsteady Discrete Adjoint Method for Shape Optimization of Three-Dimensional Multirow Turbomachinery Configurations

In turbomachinery, the steady adjoint method has been successfully used for the computation of derivatives of various objective functions with respect to design variables in gradient-based optimization. However, the continuous advances in computing power and the accuracy limitations of the steady-state assumption lead toward the transition to unsteady computational fluid dynamics (CFD) computations in the industrial design process. Previous work on unsteady adjoint for turbomachinery applications almost exclusively rely upon frequency-domain methods, for both the flow and adjoint equations. In contrast, in this paper, the development the discrete adjoint to the unsteady Reynolds-averaged Navier–Stokes (URANS) solver for three-dimensional (3D) multirow applications, in the time-domain, is presented. The adjoint equations are derived along with the adjoint to the five-stage Runge–Kutta scheme. Communication between adjacent rows is achieved by the adjoint sliding interface method. An optimization workflow that uses unsteady flow and adjoint solvers is presented and tested in two cases, with objective functions accounting for the transient flow in a turbine vane and the periodic flow in a compressor three-row setup.

Identification of Voids in Structures Based on Level Set Method and FEM

An algorithm has been developed to identify the position of voids in structures by coupling level set method and finite element method (FEM). The identification problem is transformed into a minimum problem whose objective function is defined as a least square form of displacement error. A perimeter constraint term is also added in the objective function to make the solution well posed. The level set is applied in the present algorithm to represent the position and geometry of the voids. The velocity field of level set function is obtained by analyzing the shape derivative of objective function. FEM based on Euler description is employed for solving the forward problem. The same fixed meshes adopted by the solution of forward problem are used for finite difference computation of the level set function. The procedure of this algorithm has been applied to the voids identification of two-dimensional (2D) and three-dimensional (3D) structures, the examples of single void and multiple voids are considered. The results indicate that the voids in structure can be identified effectively by the present algorithm and the algorithm is also stable to noise.

Model-Based Multiobjective Optimization Methods for Efficient Management of Subsurface Flow

Summary Multiobjective optimization (MOO), which accounts for several distinct, possibly conflicting, objectives, is expected to be capable of providing improved reservoir-management (RM) solutions for efficient oilfield development because of the overall optimization of subsurface flow. Considering the complexity and diversity of MOO problems in model-based RM, we develop three MOO methods—MOAdjoint, MOGA, and MOPSO—in this work to address various oilfield-development problems. MOAdjoint combines a weighted-sum technique with a gradient-based method for solving large-scale continuous problems that have thousands of variables. An adjoint method is used to efficiently compute the derivatives of objective functions with respect to decision variables, and a sequential quadratic-programming method is used for optimization search. MOGA is a population-based method, which combines a Pareto-ranking technique with genetic algorithm (GA) to address small-scale (discrete) problems. MOPSO is another population-based method, which combines a Pareto technique with particle-swarm optimization (PSO) for a wide spectrum of optimization problems. Their advantages and disadvantages are highlighted. To take advantage of the strengths and overcome the drawbacks of these methods, a multiscale hybrid strategy is further formulated for solving complex, large-scale optimization problems by combining these methods at various scales. An example is used to compare these methods. Results show that all three methods can yield improved solutions. MOPSO seems particularly suitable for medium-scale RM problems, mainly because of its relatively fast convergence speed and efficient recovery of the Pareto front. With a proper initial guess and a set of effective weight coefficients, MOAdjoint can most efficiently solve large-scale continuous problems, particularly if model uncertainty is considered. The multiscale hybrid strategy is able to offer the best result.