Scalar Objective Function Research Articles

A new cutting plane method for lexicographic multi-objective integer linear programming

This work presents a new cutting plane method for lexicographic multi-objective integer linear programming (LMOILP). The method uses Grossone Methodology to reformulate a LMOILP problem into one having a single scalar objective function involving Grossone, as done in Cococcioni et al. (2018) (but in that case in the absence of the integer constraints). The problem, without the integer constraints, is solved using the GrossSimplex algorithm presented in Cococcioni et al. (2018) to find a candidate optimal solution. Here a novel cutting plane is introduced, named Gross-based Objective Function Cutting Plane whenever the optimal value of the Gross-scalar objective function is Gross-fractional. When it happens to be not Gross-fractional, cutting planes are generated using the Fractional Cutting Plane, derived from fractional components of the optimal solution. Moreover, by combining them, we proposed an algorithm that we called Gross-based Cutting Plane (GCP) method. It has been proved that it finds the optimal solution of a LMOILP problem and terminates after a finite number of iterations. To speed-up the GCP, at each iteration subsequent the first one, we re-use the optimal basis of the last continuous relaxation, by running the GrossDualSimplex algorithm. This is the well-known warm-start technique, which, however, needs specific attention due to the need to solve a linear problem having a right-hand side involving Grossone-based numbers. In the experimental part, we show the efficacy of the proposed approach.

Comparison of algorithms for multi-objective optimization of radio technical device characteristics

Objectives. The selection of a method for solving multi-objective optimization problems has many practical applications in diverse fields. The present work compares the results of applying different methods to the selected classes of problems by solution quality, time consumption, and various other criteria.Methods. Five problems related to the multi-objective optimization of analog and digital filters, as well as multistep impedance-matching microwave transformers, are considered. One of the compared algorithms comprises the Third Evolution Step of Generalized Differential Evolution (GDE3) population-based algorithm for searching the full approximation of the Pareto set simultaneously, while the other three algorithms minimize the scalar objective function to find only one element of the Pareto set in a single search cycle: these comprise Multistart Pattern Search (MSPS), Multistart Sequential Quadratic Programming (MSSQP) method and Particle Swarm Optimization (PSO) algorithms.Results. The computer experiments demonstrated the capability of GDE3 to solve all considered problems. MSPS and PSO showed significantly inferior results than to GDE3 for two problems. In one problem, MSSQP could not be used to reach acceptable decisions. In the other problems, MSPS, MSSQP, and PSO reached decisions comparable with GDE3. The time consumption of the MSPS and PSO algorithms was much greater than that of GDE3 and MSSQP.Conclusions. The GDE3 algorithm may be recommended as a basic method for solving the considered problems. Algorithms minimizing scalar objective function may be used to obtain several elements of the Pareto set. It is necessary to investigate the impact of landscape features of individual quality indices and scalar objective functions on the extreme search process.

Properties of objective functions and search algorithms in multi-objective optimization problems

Objectives. A frequently used method for obtaining Pareto-optimal solutions is to minimize a selected quality index under restrictions of the other quality indices, whose values are thus preset. For a scalar objective function, the global minimum is sought that contains the restricted indices as penalty terms. However, the landscape of such a function has steep-ascent areas, which significantly complicate the search for the global minimum. This work compared the results of various heuristic algorithms in solving problems of this type. In addition, the possibility of solving such problems using the sequential quadratic programming (SQP) method, in which the restrictions are not imposed as the penalty terms, but included into the Lagrange function, was investigated.Methods. The experiments were conducted using two analytically defined objective functions and two objective functions that are encountered in problems of multi-objective optimization of characteristics of analog filters. The corresponding algorithms were realized in the MATLAB environment.Results. The only heuristic algorithm shown to obtain the optimal solutions for all the functions is the particle swarm optimization algorithm. The sequential quadratic programming (SQP) algorithm was applicable to one of the analytically defined objective functions and one of the filter optimization objective functions, as well as appearing to be significantly superior to heuristic algorithms in speed and accuracy of solutions search. However, for the other two functions, this method was found to be incapable of finding correct solutions.Conclusions. A topical problem is the estimation of the applicability of the considered methods to obtaining Pareto-optimal solutions based on preliminary analysis of properties of functions that determine the quality indices.

PyTorchDIA: a flexible, GPU-accelerated numerical approach to Difference Image Analysis

ABSTRACT We present a GPU-accelerated numerical approach for fast kernel and differential background solutions. The model image proposed in the Bramich Difference Image Analysis (DIA) algorithm is analogous to a very simple convolutional neural network (CNN), with a single convolutional filter (i.e. the kernel) and an added scalar bias (i.e. the differential background). Here, we do not solve for the discrete pixel array in the classical, analytical linear least-squares sense. Instead, by making use of PyTorch tensors (GPU compatible multidimensional matrices) and associated deep learning tools, we solve for the kernel via an inherently massively parallel optimization. By casting the DIA problem as a GPU-accelerated optimization that utilizes automatic differentiation tools, our algorithm is both flexible to the choice of scalar objective function, and can perform DIA on astronomical data sets at least an order of magnitude faster than its classical analogue. More generally, we demonstrate that tools developed for machine learning can be used to address generic data analysis and modelling problems.

Stability analysis of conically perturbed linearly constrained least-squares problems by optimizing the regularized trajectories

This paper studies linearly constrained least-square optimization problems in Hilbert spaces for which the KKT system is not necessarily available to analyze and compute the solution. The primary objective is to develop new qualitative and quantitative stability estimates for the regularization error in the conical regularization approach. To attain this goal, we associate the notion of stability with the solvability of some scalar and vector optimization problems defined in terms of the regularized trajectory on the domain space and the regularized state trajectory on the constraint space. We analyze three optimization formulations. The first formulation minimizes a scalar objective function over the regularized trajectory. The second formulation consists of vector optimizing the regularized trajectory on the domain space for a specific Bishop–Phelps cone. The third formulation results in vector optimizing the regularized state trajectory for the constraint cone. We provide numerical examples to illustrate the efficacy of the developed framework.

Simultaneous Diagonalization Using Baseband Beamforming in mmWave MU-MIMO Systems

We consider the problem of simultaneous diagonalization of Hermitian matrices for the desired and co-channel interference terms of millimeter-wave (mmWave) multi-user multiple-input multiple-output systems. The joint unitary eigenvectors and the corresponding eigenvalues are known to assist in the mathematical tractability of key performance metrics, such as outage probability, ergodic capacity, and spectral efficiency. We formulate the signal-to-interference-plus-noise ratio in a canonical quadratic form and subsume the digital baseband beamforming vectors in the weight matrices of channels at the transmitter side. Next, a real scalar objective function is defined, which quantifies the correlation loss due to joint-diagonalization. The objective function is then maximized using baseband beamforming under the hardware constraints of the mmWave system. Through simulations, the proposed beamforming algorithm is evaluated by employing several non-linear optimization sub-routines, and it is shown that the “active-set” approach results in improved summary statistics both for the correlation metric and for the time complexity. We also reflect on the effect of optimization on the channel scatterers in mmWave systems.

Designing stress for optimizing and toughening truss-like structures

Optimization of materials and structures is a crucial step in the design of man-made mechanical components for a wide field of engineering applications. It also plays a key role in mechanobiology of living systems, being involved by nature across the scales, from single-cell to tissues and organs, as a strategy to minimize metabolic cost and maximize biomechanical performances. The synergy between the continuously increasing development of high-resolution 3D printing technologies and the possibility to predict chemical and physical properties through molecular dynamics-based numerical analyses has recently contributed to boost the use of both design and topology optimization procedures. They are employed in ab initio simulations as key strategies for deciding microstructures to improve mechanical performances and, concretely, to achieve prototypes of new material components. With this in mind, we here propose to abandon the classical approach of using a single scalar objective function employed in the classical design and topology optimization strategies, to introduce multiple quantities to be minimized, identified as the differences between material yield stress and the maximum von Mises stress. After mathematically justifying the well-posedness of this unconventional choice for the case at hand, it is highlighted that the proposed strategy is based on the concept of "equalizing" a proper stress measure at any point of the body and, for this reason, it is baptized as Galilei’s optimization, in honor of the Italian scholar who somehow first wondered about the possibility of changing sizes of beams to have uniform internal forces and, in turn, minimum weight. By exploiting analytical solutions and ad hoc implementing a parametric finite element algorithm to be applied to a wide variety of solids with arbitrary complex structural geometries, including nested or hierarchically organized architectures, it is first demonstrated that the proposed optimization strategy roughly retraces principles invoked by nature to guide growth, remodeling and shaping of biomaterials. More importantly, by means of several benchmark examples, we finally show the proposed procedure might be also helpfully employed to conceive a new class of micro-structured, eventually 3D-printed materials exhibiting surprising post-elastic properties, such as high overall resilience and toughness, in particular obtaining a decrease of stress concentration and a slowing down of crack propagation as direct effects of the optimization, which de facto minimizes stress gradients wherever in the solid domain.

INverse problem of identifying a time-dependent coefficient and free boundary in heat conduction equation by using the meshless local Petrov-Galerkin (MLPG) method via moving least squares approximation

In this paper we investigated the inverse problem of identifying an unknown time-dependent coefficient and free boundary in heat conduction equation. By using the change of variable we reduced the free boundary problem into a fixed boundary problem. In direct solver problem we employed the meshless local Petrov-Galerkin (MLPG) method based on the moving least squares (MLS) approximation. Inverse reduced problem with fixed boundary is nonlinear and we formulated it as a nonlinear least-squares minimization of a scalar objective function. Minimization is performed by using of f mincon routine from MATLoptimization toolbox accomplished with the Interior - point algorithm. In order to deal with the time derivatives, a two-step time discretization method is used. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

A Stochastic Optimization Approach to Magnetometer Calibration With Gradient Estimates Using Simultaneous Perturbations

Triaxial magnetometers are indispensable part of every attitude sensing equipment for providing heading information. However, magnetometer measurements are affected by a number of error sources such as soft iron, hard iron, sensor nonorthogonality, and bias. These erroneous measurements, if not properly calibrated, deteriorate the quality of attitude estimates. In this paper, we present a new approach to calibrate triaxial magnetometer measurements by posing the calibration problem as a stochastic optimization problem. A scalar objective function is derived which involves magnetometer calibration parameters corresponding to all distortions. In order to recursively solve for optimized calibration parameters, a modified version of simultaneous perturbation stochastic approximation for the gradient is proposed for faster convergence. The proposed approach places no restrictions on the number of parameters involved and offers a complete solution by solving for all calibration parameters. The proposed approach is tested and validated using experimental data and the performance is compared against a reference approach.

Problems of a Utopia Point Setting in Transformation of Individual Objective Functions in Multi-Objective Optimization

Abstract In an optimal processes control, where the considered goals are in general observed as concurrently conflicted, a multi-objective approach fits the best. Commonly used scalarization techniques in multi-objective optimization need a transformation of the individual single-objective functions involved into a scalar multi-criteria objective function. There are many parameters which can influence the optimization results solutions, including an unreachable utopia point value. In this study, the authors compare the multi-objective problem solutions found via two ways of the individual objectives transformation with the respect to setting the utopia point. The methods are used in the area of production control in a case study for a batch production system. To find the solutions, The Weighted Sum Method with a priori articulated preferences under specific constraints as the scalar multi-objective optimization method is applied in simulation optimization.

PPLS/D: Parallel Pareto Local Search Based on Decomposition.

Pareto local search (PLS) is a basic building block in many metaheuristics for a multiobjective combinatorial optimization problem. In this paper, an enhanced PLS variant called parallel PLS based on decomposition (PPLS/D) is proposed. PPLS/D improves the efficiency of PLS using the techniques of parallel computation and problem decomposition. It decomposes the original search space into L subregions and executes L parallel processes searching in these subregions simultaneously. Inside each subregion, the PPLS/D process is guided by a unique scalar objective function. PPLS/D differs from the well-known two phase PLS in that it uses the scalar objective function to guide every move of the PLS procedure in a fine-grained manner. In the experimental studies, PPLS/D is compared against the basic PLS and a recently proposed PLS variant on the multiobjective unconstrained binary quadratic programming problems and the multiobjective traveling salesman problems with, at most, four objectives. The experimental results show that regardless of whether the initial solutions are randomly generated or generated by heuristic methods, PPLS/D always performs significantly better than the other two PLS variants.

Spatiotemporal sensitivity of systems modeled by cellular automata

Cellular automata are frequently used to model dynamics of spatial systems. This work shows how a so‐called structural adjoint sensitivity analysis may be applied to such systems. It is assumed that the cellular automaton has a continuous state, a spatiotemporal input, and is characterized by one scalar objective function specified, for example, in a given optimization or parameter optimization problem. With this analysis, it is possible to calculate efficiently a gradient of the objective function in a space of the spatiotemporal input of the automata. As an example, a model of tumor growth with an introduced spatiotemporal irradiation signal is analysed.

Using both qualitative and quantitative data in parameter identification for systems biology models

In systems biology, qualitative data are often generated, but rarely used to parameterize models. We demonstrate an approach in which qualitative and quantitative data can be combined for parameter identification. In this approach, qualitative data are converted into inequality constraints imposed on the outputs of the model. These inequalities are used along with quantitative data points to construct a single scalar objective function that accounts for both datasets. To illustrate the approach, we estimate parameters for a simple model describing Raf activation. We then apply the technique to a more elaborate model characterizing cell cycle regulation in yeast. We incorporate both quantitative time courses (561 data points) and qualitative phenotypes of 119 mutant yeast strains (1647 inequalities) to perform automated identification of 153 model parameters. We quantify parameter uncertainty using a profile likelihood approach. Our results indicate the value of combining qualitative and quantitative data to parameterize systems biology models.

SHM via topological derivative

This paper deals with the use of the topological derivative as an analysis tool for structural health monitoring, to locate the presence of flaws in a homogeneous material plate that is subject to guided elastic waves excitation. Using a numerical solver to compute the response of the system and defining a scalar objective function that measures the least squares difference between the measured and calculated signals, the topological derivative somehow describes the sensitivity of the objective function to localized perturbations of the material properties due to the presence of defects. Thus, defects are guessed to be located near the topological derivative peaks. This is somehow related to the minimization of the objective function and uses the whole physics of the problem (to compute the objective function), instead of a smaller amount of physical information, as conventional methods do. Here, we reconstruct small defects via the topological derivative by using multi-frequency synthetic data, for several representative configurations of the actuators and sensors, and several defect locations. Among these, some fairly demanding configurations are considered that are not accessible to conventional methods, such as actuators and sensors located very close to the plate boundary, and defects located beyond both elongated through-slits and elongated inclusions of a different material.

Multicriteria Robust Generalized H2 and γ0 Controllers with Application to Stabilization of a Rotor in Electromagnetic Bearings

For linear plants with unstructured or structured uncertainty of bounded norm, this paper designs Pareto optimal robust controllers in terms of linear matrix inequalities in multicriteria control problems with the generalized H2 or γ0 norms. The controller design procedure is based on optimization of a scalar objective function (Germeier convolution) and semi-definite programming. The developed theory is used to design multicriteria robust controllers in the stabilization problem for a rotor in electromagnetic bearings.

Pareto-Optimal Multi-objective Inversion of Geophysical Data

In the process of modelling geophysical properties, jointly inverting different data sets can greatly improve model results, provided that the data sets are compatible, i.e., sensitive to similar features. Such a joint inversion requires a relationship between the different data sets, which can either be analytic or structural. Classically, the joint problem is expressed as a scalar objective function that combines the misfit functions of multiple data sets and a joint term which accounts for the assumed connection between the data sets. This approach suffers from two major disadvantages: first, it can be difficult to assess the compatibility of the data sets and second, the aggregation of misfit terms introduces a weighting of the data sets. We present a pareto-optimal multi-objective joint inversion approach based on an existing genetic algorithm. The algorithm treats each data set as a separate objective, avoiding forced weighting and generating curves of the trade-off between the different objectives. These curves are analysed by their shape and evolution to evaluate data set compatibility. Furthermore, the statistical analysis of the generated solution population provides valuable estimates of model uncertainty.

Genetic algorithm for multi-parameter estimation in sorption and phase equilibria problems

ABSTRACTThe techniques of applying single and multi-objective optimization (MOO) for single/multiple parameters estimation in sorption and phase equilibria calculations were demonstrated, and it was shown that non-dominated sorting genetic algorithm with jumping genes adaptation is a useful tool for standard nonlinear regressions. Simultaneous description of vapor liquid equilibrium (VLE) and the heat of mixing (excess enthalpy) are considered a complex task in applied thermodynamics. MOO problem for simultaneous VLE and excess enthalpy prediction was formulated by (1) transforming multi-objectives into an aggregated/single scalar objective function, and (2) formulating independent objectives and solving simultaneously. It was shown that GA leads to an entire set of equally good optimal solutions known as Pareto-optimal fronts. However, simultaneous solution of MOO problem produced a wide range Pareto-optimal solution than that of the weighted sum approach. Pareto-optimal solutions are important process knowledge from which a decision-maker can opt for any set based on the applications/requirements.

Reentry Trajectory Optimization for Hypersonic Glide Vehicle with Flexible Initial Conditions

In the reentry trajectory optimization for unpowered hypersonic glide vehicles (HGVs), the initial reentry conditions (IRCs), which are crucial for flight performance, are partly flexible instead of being specified. Aiming at simultaneously determining the optimal IRCs and optimal control policy, a hybrid optimization strategy with an evaluate-during-optimize framework is proposed in this paper. The IRCs are considered as additional independent design variables, and an evaluation process with multiple performance indicators is established to determine the best IRCs. With the analytic hierarchy process (AHP)-based multicriteria decision-making (MCDM) method, a synthesized scalar objective function is set up. Adopting this objective function, feedback information from the evaluation process can be fully included in the optimization process, resulting in a unified optimization framework. The Chebyshev pseudospectral method (CPM) is then introduced as a direct method for solving the resulting trajectory optimization problem. Numerical simulation results for a generic HGV show that the proposed technique is efficient and straightforward. Moreover, this strategy has strong universality for solving optimal control problems with flexible initial conditions.

Firefly Algorithm for Finding Optimal Shapes of Electromagnetic Devices

Many real-world optimization problems have to be treated as multi-objective optimization problems. An approach, well established in recent years, is to find Pareto optimal configurations of the trial variables by detecting nondominated solutions with the help of a suitable vector optimization method. Alternatively, relying on scalar optimization methods (both stochastic or deterministic), a suitable objective function taking all objectives into account simultaneously has to be defined. Depending on the number of trial variables, a scalar objective function of that type will exhibit a considerable number of feasible local solutions besides the global one. Therefore, a useful scalar optimization strategy should be able to end up (with a high probability) in the best of all possible solutions in the given search space and additionally detect as many local solutions as possible. Some population-based stochastic methods are implicitly suited for that task; others can be enhanced to fulfill these requirements. Higher order evolution strategies have successfully been tuned for that kind of problem by introducing cluster sensitive recombination [niching higher order evolution strategy (NES)]. The firefly algorithm (FFA) mimics the behavior of fireflies, which use a kind of flashing light to communicate with other members of their species. Since the intensity of the light of a single firefly diminishes with increasing distance, the FFA is implicitly able to detect local solutions on its way to the best solution for a given scalar objective function. The FFA will be applied to the well-known Rastrigin test function and to a shielding/shunting electromagnetic problem with two and three objectives, respectively, and its results will be compared with the ones obtained with an NES.

Shape design of internal cooling passages within a turbine blade

The article concerns the optimization of the shape and location of non-circular passages cooling the blade of a gas turbine. To model the shape, four Bezier curves which form a closed profile of the passage were used. In order to match the shape of the passage to the blade profile, a technique was put forward to copy and scale the profile fragments into the component, and build the outline of the passage on the basis of them. For so-defined cooling passages, optimization calculations were carried out with a view to finding their optimal shape and location in terms of the assumed objectives. The task was solved as a multi-objective problem with the use of the Pareto method, for a cooling system composed of four and five passages. The tool employed for the optimization was the evolutionary algorithm. The article presents the impact of the population on the task convergence, and discusses the impact of different optimization objectives on the Pareto optimal solutions obtained. Due to the problem of different impacts of individual objectives on the position of the solution front which was noticed during the calculations, a two-step optimization procedure was introduced. Also, comparative optimization calculations for the scalar objective function were carried out and set up against the non-dominated solutions obtained in the Pareto approach. The optimization process resulted in a configuration of the cooling system that allows a significant reduction in the temperature of the blade and its thermal stress.