Continuous Optimization Method Research Articles

LeCaSiM: Learning Causal Structure via Inverse of M-Matrices with Adjustable Coefficients

The objective of causal discovery is to uncover the causal relationships among natural phenomena or human behaviors, thus establishing the basis for subsequent prediction and inference. Traditional ways to reveal the causal structure between variables, such as interventions or random trials, are often deemed costly or impractical. With the diversity of means of data acquisition and abundant data, approaches that directly learn causal structure from observational data are attracting more interest in academia. Unfortunately, learning a directed acyclic graph (DAG) from samples of multiple variables has been proven to be a challenging problem. Recent proposals leverage the continuous optimization method to discover the structure of the target DAG from observational data using an acyclic constraint function. Although it has an elegant mathematical form, its optimization process is prone to numerical explosion or the disappearance of high-order gradients of constraint conditions, making the convergence conditions unstable and the result deviating from the true structure. To tackle these issues, we first combine the existing work to give a series of algebraic equivalent conditions for a DAG and its corresponding brief proofs, then leverage the properties of the M-matrix to derive a new acyclic constraint function. Based on the utilization of this function and linear structural equation models, we propose a novel causal discovery algorithm called LeCaSiM, which employs continuous optimization. Our algorithm exploits the spectral radius of the adjacency matrix to dynamically adjust the coefficients of the matrix polynomial, effectively solving the above problems. We conduct extensive experiments on both synthetic and real-world datasets, the results show our algorithm outperforms the existing algorithms on both computation time and accuracy under the same settings.

COMBSS: best subset selection via continuous optimization

The problem of best subset selection in linear regression is considered with the aim to find a fixed size subset of features that best fits the response. This is particularly challenging when the total available number of features is very large compared to the number of data samples. Existing optimal methods for solving this problem tend to be slow while fast methods tend to have low accuracy. Ideally, new methods perform best subset selection faster than existing optimal methods but with comparable accuracy, or, being more accurate than methods of comparable computational speed. Here, we propose a novel continuous optimization method that identifies a subset solution path, a small set of models of varying size, that consists of candidates for the single best subset of features, that is optimal in a specific sense in linear regression. Our method turns out to be fast, making the best subset selection possible when the number of features is well in excess of thousands. Because of the outstanding overall performance, framing the best subset selection challenge as a continuous optimization problem opens new research directions for feature extraction for a large variety of regression models.

Coupling of bond-based peridynamics and continuous density-based topology optimization methods for effective design of three-dimensional structures with discontinuities

AbstractThis study proposes continuous density-based three-dimensional topology optimization (TO) approaches developed by coupling the peridynamic theory (PD) with optimality criteria (OC) and proportional approach (PROP). These frameworks, abbreviated as PD-OC-TO and PD-PROP-TO, can be practically utilized to enhance the fracture toughness of the structures during the optimization process by taking critical regions into account as pre-defined cracks. Breaking the non-local interactions (bonds) between relevant PD particles enables us to readily model cracks. Utilizing this advantage, we solve several benchmark optimization problems including different numbers, positions, and alignments of the cracks. The major differences between the proposed methods are examined by comparing optimum topologies for various cracked scenarios. Moreover, the mechanical behaviour of the optimized structures is investigated under dynamic loads to prove the significant improvements achieved by the present approach in the final designs. The results of dynamic analyses reveal the viability of both PD-TO methods for increasing the fracture toughness of the structure in the optimization stage. Overall, the proposed approach is confirmed as a superior design and optimization tool for future engineering structures. Graphical abstract

Topology Design of 3D Printing Continuous Fiber‐Reinforced Structure Considering Strength and Non‐Equidistant Fiber

The utilization of continuous fiber‐reinforced composites (CFRC) technology in 3D printing opens up new avenues for designing composite products. This paper introduces a novel parallel topology optimization framework tailored for CFRC 3D printing. The framework incorporates strength constraints and uses topology, fiber volume fraction, and fiber orientation as essential design variables. A fiber material interpolation model for multiple design variables is proposed based on the law of mixtures for composites. Bidirectional evolutionary structural optimization and solid isotropic material with penalization are used for optimization of topology and fiber volume fraction, respectively, while the fiber orientation herein is determined using the principal stress method. Strength constraint is considered according to Tsai–Hill criterion. To match manufacturing process demands, a nonequidistant continuous fiber path optimization method based on Hermite interpolation function is introduced. The optimization framework introduced herein is successfully used for optimal design of L‐shaped beam, cantilever beam, and Michell beam. It is concluded that the design space of composite structure can be further broadened using the proposed method. To validate the viability of the proposed optimization method and fiber path design approach, the optimization results are printed using a custom‐developed continuous fiber 3D printer.

Study on continuous adjoint-based optimization method for internal cooling structures in turbine blades with conjugate heat transfer

A continuous adjoint-based shape optimization system is presented in this paper for conjugate heat transfer (CHT) problems and is applied to the optimization for internal cooling structures of gas turbine blades. With the consideration for the variation of heat conduction equation in the solid regions, the adjoint equations and adjoint boundary conditions in different regions are derived. The adjoint energy correction equation is also derived successfully based on the SIMPLE algorithm, and the adjoint CHT solver is implemented by imposing the values of adjoint energy at the interfaces. Two cases of full 3D optimization for the C3X hollow blade and cooling channel with pin fins are conducted by using the optimization system. In both cases, their flow and heat transfer characteristics are optimized and analyzed in detail. This study demonstrates the potential advantages and basic capabilities of the continuous adjoint method in dealing with CHT optimization problems.

Optimal design of strain sensor placement for distributed static load determination

Abstract In many applications it is desirable to inverse-calculate the distributed loading on a structure using a limited number of sensors. Yet, the calculated loads can be extremely sensitive to the placement of these sensors. In the case of predicting point loading applied at a known location, best results are typically achieved when one sensor is collocated with the force. However, the extension of this rule to distributed loading remains uncertain, and even simple sensor system design scenarios often require the designer to directly optimize the sensor placements using a numerical model. In an effort to provide designers with guidance, we identify optimal sensor configurations for predicting static distributed loads on beams with classical boundary conditions. An influence coefficient method, wherein the strain is related linearly to the static load, is used to estimate the applied forces. The loading distribution on the structure is assumed to be either a piece-wise linearly-distributed load or a uniformly-distributed load, allowing for distributed loads to be estimated using the magnitudes of a small number of control points. Given the simplicity of the beam structure, the equations of the influence coefficient method are derived analytically, which allows for the sensor placement to be specified using continuous optimization methods. The condition number of the influence coefficient matrix is used as a surrogate for error during optimization. ‘Rules of thumb’ for sensor placement are presented based on the optimization results. Results show that the optimal and rule-of-thumb sensor configurations are more resistant to input noise than naïve configurations, with the rule-of-thumb configurations yielding similar force predictions relative to the optimal configurations. We expect the rules of thumb to be useful guidelines for engineers designing tests on beam-like structures such as aircraft wings or marine propellers where the inverse calculation of distributed loads is of interest.

Concurrent optimization method of principal stress orientation interpolated continuous fiber angle (PSO-CFAO) and structural topology

Continuous fiber-reinforced polymers (CFRPs) have been widely applied in aerospace and other fields due to their excellent mechanical properties, which highly depends on the material distribution and fiber orientations. The designability of CFRP structures and fiber distributions provides an opportunity for achieving better physical properties through optimization. However, local optima and the dependence of initial fiber angle variables make the concurrent optimization of fiber orientation and topology a challenging problem. In this paper, the principal stress orientation interpolated continuous fiber angle optimization (PSO-CFAO) method combined with the independent continuous mapping (ICM) method is proposed to realize the design of CFRP structures with a clear macroscopic topology and microscopic fiber distribution. A sigmoid function is applied to interpolate the fiber angle variables by the principal stress orientation. The fiber angle variables are modified and a continuous fiber design is obtained during the iteration process, which reduces the possibility of a local optimum. Several examples are provided to prove the effectiveness and stability of the proposed method, and the expected results are acquired for different initial fiber angles, material parameters and mesh densities. The proposed method provides guidance for the design of CFRP structures and the planning of fiber laying paths.

Newton’s Method for Global Free Flight Trajectory Optimization

Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton’s method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case.

Optimization of formworks shoring location as a continuous optimization problem

For the construction of reinforced concrete buildings, partial striking of props and formworks a few days after casting is frequently employed, confirming its safety based on structural calculations or analyses to improve the construction productivity. Three-dimensional finite element (FE) models are sometimes used for these kinds of structural analyses to obtain precise results. However, considerable effort is required to determine the remaining prop arrangement by trial and error via FE analysis because of time-consuming operations. This study presents continuous optimization methods for the determination of the arrangement of remaining props that can speedily provide a single valid solution based on design sensitivity; the solution guarantees local optimality. Continuous optimizations employ a function that transforms the continuous design variables into the number of remaining props at each candidate location that is inherently represented by integer values. The optimization problems have been formulated in two forms: one is to maximize the safety of the structure, and the other is to minimize the number of remaining props. These optimization methods provided valid local optimal solutions that could reduce the amount of material required for formwork and shoring by almost half within a realistic calculation time. In addition, compared with multi-objective discrete optimization by a genetic algorithm, the first method provided the optimal solution with less than half the number of FE analyses of discrete method, and the second method provided a more efficient solution than the discrete method with a total of 10,000 individuals.

Topology Optimization of Minimum Compliance for Continuous Fiber-Reinforced Polymer

Continuous fiber-reinforced polymer (CFRP) has been paid more attention to due to its advantages of high strength, high stiffness, and lightweight. However, it is challenging for topology optimization to achieve CRRP structure design. A topology optimization (TO) model with minimizing structural compliance under volume fraction constraint is established based on the independent continuous mapping (ICM) method. The continuous fiber angle optimization (CFAO) method solves the fiber orientation. Meanwhile, the local optimal solution problem caused by the CFAO method is greatly avoided by the direction of principal stress. The effectiveness of the method is verified by classic numerical examples. This paper has a significant reference for the TO design of CFRP structure.

A Dynamic Continuous Constrained Phase Factor Graph Optimization Method for GNSS Kinematic Precise Point Positioning

Carrier phase measurements play a more critical role in achieving precise positioning for lower noise and insensitivity to multipath error over code (pseudorange) measurements. Real-time kinematic (RTK) is the most used precise positioning technique. However, the application costs are increased due to the demands for differential base stations. The undifferenced precise point positioning (PPP) does not require differential station assistance but is typically for static use. This paper proposes a dynamic continuous constrained phase (DCCP) method based on factor graph optimization for kinematic positioning without differential stations, even in the presence of cycle slips. The precise velocity estimated via integral doppler and the time correlation of phase ambiguity are regarded as the probability constraints in the factor graph. A real-world experiment was carried out to verify the feasibility of the DCCP method. Experimental results show that the proposed method had a better performance than the kinematic PPP method and window carrier phase (WCP) optimization method with 0.034m standard deviation and 1.338m mean error. It is our hope that this approach provides new insight into the study of other time-correlated state estimation problems.

Robust transformed [formula omitted] metric for fluorescence molecular tomography

Background and Objective: Fluorescence molecular tomography (FMT) is a non-invasive molecular imaging modality that can be used to observe the three-dimensional distribution of fluorescent probes in vivo. FMT is a promising imaging technique in clinical and preclinical research that has attracted significant attention. Numerous regularization based reconstruction algorithms have been proposed. However, traditional algorithms that use the squared l2-norm distance usually exaggerate the influence of noise and measurement and calculation errors, and their robustness cannot be guaranteed.Methods: In this study, we propose a novel robust transformed l1 (TL1) metric that interpolates l0 and l1 norms through a nonnegative parameter α∈(0,+∞). The TL1 metric looks like the lp-norm with p∈(0,1). These are markedly different because TL1 metric has two properties, boundedness and Lipschitz-continuity, which make the TL1 criterion suitable distance metric, particularly for robustness, owing to its stronger noise suppression. Subsequently, we apply the proposed metric to FMT and build a robust model to reduce the influence of noise. The nonconvexity of the proposed model made direct optimization difficult, and a continuous optimization method was developed to solve the model. The problem was converted into a difference in convex programming problem for the TL1 metric (DCATL1), and the corresponding algorithm converged linearly.Results: Various numerical simulations and in vivo bead-implanted mouse experiments were conducted to verify the performance of the proposed method. The experimental results show that the DCATL1 algorithm is more robust than the state-of-the-art approaches and achieves better source localization and morphology recovery.Conclusions: The in vivo experiments showed that DCATL1 can be used to visualize the distribution of fluorescent probes inside biological tissues and promote preclinical application in small animals, demonstrating the feasibility and effectiveness of the proposed method.

Topology optimization of bilayer thermal scattering cloak based on CMA-ES

A topology optimization method for the design of thermal cloak is proposed. The thermal scattering theory is introduced to pre-define the initial topology of the thermal invisible cloak as two layers to reduce the number of design variables. In bilayer concentric rings, insulating and high thermal conductivity materials are used to mask heat flow and correct thermal properties. Topology optimization based on the CMA-ES strategy is applied to quickly obtain the best configuration of available materials. The influence of the thermal properties of the material on the optimal configuration and the deflection angle of the thermal flux at the boundary surface has been investigated. Numerical simulations were performed to compare the function of the optimal cloak with that of the ideal metamaterial and the single insulating layer. The results show that despite its simple structure, the bilayer scattering cloak exhibits excellent cloaking performance under different thermal boundary conditions. In this work, the perfect scattering elimination is achieved by adjusting the layer thickness ratio of the cloak with the initial defined structure rather than seeking the optimal thermal conductivity of material, so as to avoid the problem of mismatch between the actual and ideal parameters. Compared to continuous optimization method, the pre-defined bilayer structure is easy to manufacture. Our scheme introduces thermal scattering theory to pre-set the initial structure of the topology optimization method, which provides a general, convenient, and efficient means for the design and application of thermally functional materials.

An Effective Power Dispatch of Photovoltaic Generators in DC Networks via the Antlion Optimizer

This paper studies the problem regarding the optimal power dispatch of photovoltaic (PV) distributed generators (DGs) in Direct Current (DC) grid-connected and standalone networks. The mathematical model employed considers the reduction of operating costs, energy losses, and CO2 emissions as objective functions, and it integrates all technical and operating constraints implied by DC grids in a scenario of variable PV generation and power demand. As a solution methodology, a master–slave strategy was proposed, whose master stage employs Antlion Optimizer (ALO) for identifying the values of power to be dispatched by each PV-DG installed in the grid, whereas the slave stage uses a matrix hourly power flow method based on successive approximations to evaluate the objective functions and constraints associated with each solution proposed within the iterative process of the ALO. Two test scenarios were considered: a grid-connected network that considers the operating characteristics of the city of Medellín, Antioquia, and a standalone network that uses data from the municipality of Capurganá, Chocó, both of them located in Colombia. As comparison methods, five continuous optimization methods were used which were proposed in the specialized literature to solve optimal power flow problems in DC grids: the crow search algorithm, the particle swarm optimization algorithm, the multiverse optimization algorithm, the salp swarm algorithm, and the vortex search algorithm. The effectiveness of the proposed method was evaluated in terms of the solution, its repeatability, and its processing times, and it obtained the best results with respect to the comparison methods for both grid types. The simulation results obtained for both test systems evidenced that the proposed methodology obtained the best results with regard to the solution, with short processing times for all of the objective functions analyzed.

Cathode Shape Design for Steady-State Electrochemical Machining

The inverse or cathode shape design problem of electrochemical machining (ECM) deals with the computation of the shape of the tool cathode required for producing a workpiece anode of a desired shape. This work applied the complex variable method and the continuous adjoint-based shape optimization method to solve the steady-state cathode shape design problem with anode shapes of different smoothnesses. An exact solution to the cathode shape design problem is proven to exist only in cases when the function describing the anode shape is analytic. The solution’s physical realizability is shown to depend on the aspect ratio of features on the anode surface and the width of the standard equilibrium front gap. In cases where an exact and physically realizable cathode shape exists, the continuous adjoint-based shape optimization method is shown to produce accurate numerical solutions; otherwise, the method produces cathode shapes with singularities. For the latter cases, the work demonstrates how perimeter regularization can be applied to compute smooth approximate cathode shapes suitable for producing workpieces within the range of manufacturing tolerance.

A Generator of Multiextremal Test Classes With Known Solutions for Black-Box-Constrained Global Optimization

A generator of classes of multidimensional test problems for benchmarking continuous constrained global optimization methods is proposed. It is based on the generator of test classes for global optimization proposed in 2003 by Gaviano, Kvasov, Lera, and Sergeyev and extends the previous generation procedure from the box-constrained case to the case of nonlinear constraints. The user has the possibility to fix the difficulty of tests in an intuitive way by choosing several types of constraints. A detailed information (including the global solution) for each of 100 problems in each generated class is provided to the user. The generator is particularly suited for testing black-box optimization algorithms that normally address low or medium dimensional problems with hard to evaluate objective functions.

D’ya Like DAGs? A Survey on Structure Learning and Causal Discovery

Causal reasoning is a crucial part of science and human intelligence. In order to discover causal relationships from data, we need structure discovery methods. We provide a review of background theory and a survey of methods for structure discovery. We primarily focus on modern, continuous optimization methods, and provide reference to further resources such as benchmark datasets and software packages. Finally, we discuss the assumptive leap required to take us from structure to causality.

Optimizing human hand gestures for AI-systems

Humans interact more and more with systems containing AI components. In this work, we focus on hand gestures such as handwriting and sketches serving as inputs to such systems. They are represented as a trajectory, i.e. sequence of points, that is altered to improve interaction with an AI model while keeping the model fixed. Optimized inputs are accompanied by instructions on how to create them. We aim to cut on effort for humans and recognition errors while limiting changes to original inputs. We derive multiple objectives and measures and propose continuous and discrete optimization methods embracing the AI model to improve samples in an iterative fashion by removing, shifting and reordering points of the gesture trajectory. Our quantitative and qualitative evaluation shows that mimicking generated proposals that differ only modestly from the original ones leads to lower error rates and requires less effort. Furthermore, our work can be easily adjusted for sketch abstraction improving on prior work.

Derivation of a continuous adjoint topology optimization method applied on heat transfer for incompressible flow

With the development of additive manufacturing, the freedom in developing complex cooling structures has been enlarged greatly. Topology optimization is one of the methods that can give out cooling structures beyond the limits of geometric parameters. In this research, an algorithm of topology optimization with heat transfer is derived. The detailed optimization procedure is presented in this work. The basic assumptions of this optimization method are steady state, incompressible and frozen turbulent flow with constant physical properties. The governing equations, optimization sensitivity formula and boundary conditions are presented, which are essential to the implementation of this method. A detailed derivation process is included. This method is applied in a 2-D flow domain using an open-source platform named OpenFOAM.

Conflict-Directed Diverse Planning for Logic-Geometric Programming

Robots operating in the real world must combine task planning for reasoning about what to do with motion planning for reasoning about how to do it -- this is known as task and motion planning. One promising approach for task and motion planning is Logic Geometric Programming (LGP) which integrates a logical layer and a geometric layer in an optimization formulation. The logical layer describes feasible high-level actions at an abstract symbolic level, while the geometric layer uses continuous optimization methods to reason about motion trajectories with geometric constraints. In this paper we propose a new approach for solving task and motion planning problems in the LGP formulation, that leverages state-of-the-art diverse planning at the logical layer to explore the space of feasible logical plans, and minimizes the number of optimization problems to be solved on the continuous geometric layer. To this end, geometric infeasibility is fed back into planning by identifying prefix conflicts and incorporating this back into the planner through a novel multi-prefix forbidding compilation. We further leverage diverse planning with a new novelty criteria for selecting candidate plans based on the prefix novelty, and a metareasoning approach which attempts to extract only useful conflicts by leveraging the information that is gathered in the course of solving the given problem.