Deterministic Global Optimization Research Articles

Multiobjective optimization by using cutting angle methods and hypervolume criterion

We translate a multiobjective optimization problem into a single objective Lipschitz problem by using the hypervolume criterion of the Pareto set. Deterministic global optimization methods allow one to track the whole Pareto front rather than converging to a single non-dominated solution. We augmented an efficient hypervolume computation technique with hypervolume increment strategy, and established Lipschitzianity of the resulting objective function. We used two deterministic Lipschitz optimization methods together with the hypervolume objective and benchmarked them against some state-of-the-art alternative multiobjective optimization methods, establishing the competitiveness of the proposed approach.

Global optimization for large‐scale water network synthesis based on dynamic partition and adaptive bound tightening

AbstractThe synthesis of large‐scale integrated water networks is typically formulated as nonconvex mixed‐integer quadratic constrained programming (MIQCP) or QCP problems. With the complexity arising from bilinear terms in modeling mass flows of contaminants and binary variables representing the presence of units or streams, numerous local optima exist, thus presenting a significant optimization challenge. This study introduces a deterministic global optimization algorithm based on mixed‐integer programming (MIP) to tackle such problems. The approach involves dynamically strengthening the relaxed problems to converge towards the original problems. A simultaneous partition strategy is proposed combining locally uniform division with dynamic partitioned variables choosing. Furthermore, several adaptive bound contraction schemes are introduced to efficiently manage the size of the relaxed problems, assisting in accelerating the solution process. The algorithm's effectiveness and robustness are demonstrated with a large test set, showing superior performance compared to commercial solvers specifically on MIQCP problems.

Automatic source code generation for deterministic global optimization with parallel architectures

Trends over the past two decades indicate that much of the performance gains of commercial optimization solvers is due to improvements in x86 hardware. To continue making progress, it is critical to consider alternative/specialized massively parallel computing architectures. In this work, we detail the development of an open-source source code transformation approach built using Symbolics.jl to construct McCormick-based relaxations of functions that enables their effective parallelized evaluation. We then apply this approach in a novel parallelized branch-and-bound routine that offloads lower- and upper-bounding problems to a GPU. The effectiveness of this new approach is demonstrated on three nonconvex problems of interest, where it yields convergence time improvements of 22–118x compared to an equivalent serial CPU implementation and in two cases outperforms vanilla branch-and-bound versions of existing state-of-the-art solvers that use tighter bounding techniques. This work exemplifies how deterministic global optimizers using alternative hardware architectures can compete with—or eventually outclass—even the most powerful serial CPU implementations, and to the best of the authors' knowledge, represents the first successful demonstration of deterministic global optimization using a GPU.

A new filled function for global optimization and its deterministic efficiency in solving the facility layout/location problem

Although many real-world problems can be modelled as optimization problems, the multimodal nature of such problems and the fact that traditional algorithms are often trapped within local minima make it very difficult to solve these problems. The filled function method is one of the highly effective deterministic global optimization techniques that can overcome such difficulties. However, it is known that the use of non-differentiable and parameter-sensitive filled functions in the solution of global optimization problems could cause high time and computational costs. In this study, a continuously differentiable filled function is proposed to overcome such disadvantages. In addition, for the first time in the literature, the solution of the facility layout problem has been introduced using the filled function method. Moreover, it has been shown in the numerical experiment that, even with more than twenty facilities, this problem can be solved within a very satisfactory time and computational cost.

A branch-and-bound algorithm with growing datasets for large-scale parameter estimation

The solution of nonconvex parameter estimation problems with deterministic global optimization methods is desirable but challenging, especially if large measurement datasets are considered. We propose to exploit the structure of this class of optimization problems to enable their solution with the spatial branch-and-bound algorithm. In detail, we start with a reduced dataset in the root node and progressively augment it, converging to the full dataset. We show for nonlinear programs (NLPs) that our algorithm converges to the global solution of the original problem considering the full dataset. The implementation of the algorithm extends our open-source solver MAiNGO. A numerical case study with a mixed-integer nonlinear program (MINLP) from chemical engineering and a dynamic optimization problem from biochemistry both using noise-free measurement data emphasizes the potential for savings of computational effort with our proposed approach.

An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron.

Technoeconomic assessment of solar technologies for the hybridization of industrial process heat systems using deterministic global dynamic optimization

The industrial process heat (IPH) sector provides an opportunity to replace terawatts of fossil fuel energy with solar alternatives, such as parabolic trough collectors (PTC) or photovoltaics (PV) with resistive heating and energy storage (ES) for off-peak deployment. PTCs offer significantly higher thermal efficiency, but have stagnated in cost and efficiency improvements over the past decade. In contrast, PVs have seen significant improvements in cost and efficiency. Additionally, the energy generated by PV can be stored electrochemically in batteries. Current optimization-based approaches for design and investment decision making lack guarantees of global optimality that are necessary to ensure the best-possible solutions are obtained for these general nonconvex dynamical models. We present dynamical models for the formal deterministic global optimization-based design and technoeconomic assessment of solar systems to hybridize IPH and reduce natural gas combustion that serve as a general and flexible framework. We compare hybridization strategies of PTC with thermal ES, PV with thermal ES, and PV with battery ES and explore the impacts of natural gas pricing and process scale. For the low- to medium-temperature IPH, PTC with thermal ES is currently at least four times more economically favorable than alternative technologies regardless of location or process scale.

Exploring synchronization and lift suppression in fluid flow around vibrating cylinder: a parallel CFD and global optimization investigation

In this study, we employ a parallel Computational Fluid Dynamics (CFD) code integrated with the VTDIRECT95 algorithm, a parallel deterministic global optimization method, to conduct global optimization for an oscillating circular cylinder. We conduct numerical simulations for the flow at a Reynolds number of 500 within the parameter range of 0.1≤Ax≤0.3 and 0.5fst≤fex≤2.5fst where Ax represents the inline oscillating amplitude, fex denotes the forcing oscillation frequency, and fst corresponds to the frequency of a stationary cylinder. To enhance computational efficiency, a combination of VTdirect and a CFD solver is utilized to efficiently identify the synchronization region, thereby reducing computational resources. The results reveal a significant reduction in the lift coefficient within the synchronized region compared to unsynchronized regimes. Furthermore, the study delves into the underlying flow physics behind synchronization and lift suppression. By synchronizing the shedding of vortices, their detrimental effects are nullified, resulting in a reduction in lift. Moreover, the research examines the influence of three-dimensional (3-D) flow by comparing 2-D and 3-D simulations at two different Reynolds numbers. It demonstrates that accounting for 3-D effects yields more accurate predictions of fluid behavior. Synchronization maps and root mean square (rms) lift coefficient plots illustrate the impact of Reynolds number and movement frequency on lift suppression. The findings indicate that achieving synchronization in 3-D flow necessitates stronger amplitudes and higher frequencies. At higher Reynolds numbers, the wake structures become unstable, leading to intricate vortical patterns. Consequently, the synchronization curve shifts towards higher amplitudes and frequencies in 3-D simulations. Understanding these phenomena is vital for reducing lift force in practical applications. This research significantly contributes to knowledge regarding synchronization and lift suppression in fluid flow around vibrating cylinders.

Towards global parameter estimation exploiting reduced data sets

We focus on deterministic global optimization (DGO) for nonconvex parameter estimation problems. Realistic and accurate solutions often require a fitting against very large measurement data sets, resulting in intractable size for DGO. Thus, we aim at accelerating the branch-and-bound algorithm by using reduced data sets for constructing valid bounds. We focus on fitting the equation of state for propane, which is of high interest for the chemical industry. The resulting estimation problem is a challenging nonconvex mixed-integer nonlinear optimization problem. We investigate the validity of using reduced data sets by comparing how the lower and upper bounds change when replacing the full data set with different reduced data sets. We account for regions with high and low quality fits by considering the results for the whole feasible region and 100 different subregions. Our results indicate that both regions containing solution candidates and regions containing only low quality fits can be identified based on reduced data sets. Moreover, we observe that the average CPU time per branch-and-bound iteration typically decreases if reduced data sets are used.

A rigorous deterministic global optimization approach for the derivation of secondary information in digital maps

We derive a generic system that constructs an optimization model for an emergency stop scenario on the highway, based on map data from high definition maps that are used in Advanced Driver Assistance Systems (ADAS) and in Highly Automated Driving (HAD). New additional situative and scenario-based information is computed by applying a global maximization approach to the model. For this purpose, we develop two new rigorous and deterministic branch-and-bound algorithms that both determine the certified global optimal value up to a predefined tolerance. The underlying interval optimization algorithm, which uses first-order techniques, is enhanced by one of two second-order methods that are applied for specifically selected intervals. We investigate two approaches that either compute a concave overestimator for the objective function or approximate the function with a quadratic polynomial using Taylor expansion. We show the limits of interval arithmetic in our problem, especially for the interval versions of the derivatives, and present a local linearization of the curve data that improves the results significantly. The presented novel method for deriving secondary information is compared to state of the art methods on two exemplary and for the automotive context representative scenarios to show the advantages of our approach.

Collaborative granular sieving: A deterministic multievolutionary algorithm for multimodal optimization problems

Evolutionary algorithms (EAs) that integrate niching techniques are among the most effective methods for multimodal optimization problems. However, most algorithmic contributions are based on empirical performance observations rather than rigorous mathematical convergence support; this makes most existing methods parameter sensitive. Inspired by a recently proposed deterministic global optimization method, granular sieving (GrS), an extended global optimization method named collaborative GrS (Co-GrS) and a novel deterministic multi-EA design framework are proposed in this paper. The innovations are threefold. (1) Existing EAs are stochastic methods, and this paper introduces the principle of deterministic global optimization into EA for the first time in the literature. (2) A deterministic multi-EA framework is designed and implemented in the paper; from the perspective of population evolution, an easy-to-operate survival-of-the-fittest strategy based on mathematical principles is established in Co-GrS. (3) Unlike existing stochastic EAs, where the reproducibility of optimal solutions is achieved in a statistical sense, Co-GrS does not involve random parameters, and it automatically runs the algorithm only once with pre-set fixed parameters to find all optimal solutions. The experimental results demonstrate the effectiveness and competitiveness of our method compared to 16 state-of-the-art multimodal algorithms on the CEC’2013 benchmark suite.

Convex and concave envelopes of artificial neural network activation functions for deterministic global optimization

Convex and concave envelopes of artificial neural network activation functions for deterministic global optimization

Optimal therapy design with tumor microenvironment normalization

AbstractTumor microenvironment (TME) normalization improves efficacy by increasing anticancer nanocarrier delivery by restoring transvascular pressure gradients that induce convection. However, transport depends on TME biophysics, normalization dose, and nanocarrier size. With increased understanding, we could use computation to personalize normalization amount and nanocarrier size. Here, we use deterministic global dynamic optimization with novel bounding routines to validate mechanistic models against in vivo data. We find that normalization with dexamethasone increases the maximum transvascular convection rate of nanocarriers by 48‐fold, the tumor volume fraction with convection by 61%, and the total amount of convection by 360%. Nonetheless, 22% of the tumor still lacks convection. These findings underscore both the effectiveness and limits of normalization. Using artificial neural network surrogate modeling, we demonstrate the feasibility of rapidly determining the dexamethasone dose and nanocarrier size to maximize accumulation. Thus, this digital testbed quantifies transport and performs therapy design.

Simultaneous deterministic global flowsheet optimization and heat integration: Comparison of formulations

Heat integration can be considered in flowsheet optimization by including the minimum utility demand from pinch analysis in the objective and constraints. This often results in better process performance than conducting these steps separately. However, it comes with increased computational cost, especially for global optimization. This cost depends both on the problem formulation and on the solver. In this work, we compare several existing and new smooth, nonsmooth, and mixed-integer formulations. Furthermore, we test different choices of optimization variables and constraints reaching from full- to reduced-space formulations. In the reduced-space formulations, heat integration can be included with few, one, or even zero additional variables beyond the pure flowsheet optimization problem. For the considered case studies, this can significantly reduce the solution time of various global solvers, in particular for our open-source solver MAiNGO. Depending on the case study and solver, either nonsmooth or mixed-integer formulations are the fastest to solve.

Global dynamic optimization with Hammerstein–Wiener models embedded

Hammerstein–Wiener models constitute a significant class of block-structured dynamic models, as they approximate process nonlinearities on the basis of input–output data without requiring identification of a full nonlinear process model. Optimization problems with Hammerstein–Wiener models embedded are nonconvex, and thus local optimization methods may obtain suboptimal solutions. In this work, we develop a deterministic global optimization strategy that exploits the specific structure of Hammerstein–Wiener models to extend existing theory on global optimization of systems with linear dynamics. At first, we discuss alternative formulations of the dynamic optimization problem with Hammerstein–Wiener models embedded, demonstrating that careful selection of the optimization variables of the problem can offer significant numerical advantages to the solution approach. Then, we develop convex relaxations for the proposed optimization problem and discuss implementation aspects to obtain the global solution focusing on a control parametrization technique. Finally, we apply our optimization strategy to case studies comprising both offline and online dynamic optimization problems. The results confirm an improved computational performance of the proposed solution approach over alternative options not exploiting the linear dynamics for all considered examples. They also underline the tractability of deterministic global dynamic optimization when using few control intervals in online applications like nonlinear model predictive control.

Globally optimal shape and spin pole determination with light-curve inversion

ABSTRACT Light-curve inversion is an established technique in determining the shape and spin states of an asteroid. However, the front part of the processing pipeline, which recovers the spin pole and area of each facet, is a non-convex optimization problem. Hence, any local iterative optimization scheme can only promise a locally optimal solution. Apart from the obvious downsides of getting a non-optimal solution and the need for an initialization scheme, another major implication is that it creates an ambiguous scenario – which is to be blamed for the remaining residual? The inaccuracy of the modelling, the integrity of the data, or the non-global algorithm? We address the last uncertainty in this paper by embedding the spin pole and area vector determination module in a deterministic global optimization framework. To the best of our knowledge, this is the first attempt to solve these parameters globally. Specifically, given calibrated light-curve data, a scattering model for the object, and spin period, our method outputs the globally optimal spin pole and area vector solutions. One theoretical contribution of this paper is the introduction of a lower bound error function that is derived based on (1) the geometric relationship between the incident and scattered light on a surface and (2) the uncertainty of the gap between the observed and estimated brightness at a particular epoch in a light curve. We validated our method’s ability in achieving global minimum with both simulated and real light-curve data. We also tested our method on the real light curves of four asteroids.

The application of nonlinear programming on ration formulation for dairy cattle

The objective of this study was to compare the application of iterative linear programming (iteLP), sequential quadratic programming (SQP), and mixed-integer nonlinear programming-based deterministic global optimization (MINLP_DGO) on ration formulation for dairy cattle based on Nutrient Requirements of Dairy Cattle (NRC, 2001). Least-cost diets were formulated for lactating cows, dry cows, and heifers. Nutrient requirements including energy, protein, and minerals, along with other limitations on dry matter intake, neutral detergent fiber, and fat were considered as constraints. Five hundred simulations were conducted, with each simulation randomly selecting 3 roughages and 5 concentrates from the feed table in NRC (2001) as the feed resource for each of 3 animal groups. Among the 500 simulations for lactating cows, 57, 45, and 21 simulations did not yield a feasible solution when using iteLP, SQP, and MINLP_DGO, respectively. All the simulations for dry cows and heifers were feasible when using SQP and MINLP_DGO, but 49 and 11 infeasible simulations occurred when using iteLP for dry cows and heifers, respectively. The average ration costs per animal per day of the feasible solutions obtained by iteLP, SQP, and MINLP_DGO were $4.78 (±0.71), $4.45 (±0.65), and $4.44 (±0.65) for lactating cows; $2.39 (±0.52), $1.48 (±0.26), and $1.48 (±0.26) for dry cows; and $0.98 (±0.72), $0.97 (±0.15), and $0.91 (±0.14) for heifers, respectively. The average computation time of iteLP, SQP, and MINLP_DGO were 0.59 (±1.87) s, 1.15 (±0.62) s, and 58.69 (±68.45) s for lactating cows; 0.041 (±0.070) s, 0.76 (±0.37) s, and 14.84 (±39.09) s for dry cows; and 1.60 (±2.90) s, 0.51 (±0.19) s, and 16.45 (±45.56) s for heifers, respectively. In conclusion, iteLP had limited capability of formulating least-cost diets when nonlinearity existed in the constraints. Both SQP and MINLP_DGO handled the nonlinear constraints well, with SQP being faster, whereas MINLP_DGO was able to return a feasible solution under some situations where SQP could not.

Flexible operation of modular electrochemical CO2 reduction processes

Electrochemical CO2 reduction (eCO2R) is an emerging technology that is capable of producing various organic chemicals from CO2, but its high electricity cost is a big economic obstacle. One solution to reduce the cumulative electricity cost is demand side management, i.e., to adjust the power load based on time-variant electricity prices. However, varying the power load of CO2-electrolyzers often leads to changes in Faraday efficiency towards target components and thereby influences the product composition. Such deviations from the target product composition may be undesired for downstream processes. We tackle this challenge by proposing a flexible operating scheme for a modular eCO2R process. We formulate the economically optimal operation of an eCO2R process with multiple electrolyzer stacks as a parallel-machine scheduling problem. Adjusting the power load of each sub-process properly, we can save electricity costs while the desired product composition is met at any time. We apply an algorithm based on wavelet transform to solve the resulting large-scale nonlinear scheduling problem in tractable time. We solve each optimization problem with a deterministic global optimization software MAiNGO. We examine flexible operation of a modular eCO2R process for syngas production. The case studies show that the modular structure enables savings in the cumulative electricity cost of the eCO2R process via flexible operation while deviations in the syngas composition could be reduced. Also, the maximum ramping speed of the entire process is found to be a key parameter that strongly influences the cost saving.

Precise control of water and wastewater treatment systems with non-ideal heterogeneous mixing models and high-fidelity sensing

Non-ideal heterogeneous mixing models are developed and incorporated within advanced closed-loop control strategies utilizing high-resolution sensing to maximize the resiliency and minimize the energy consumption of water treatment processes with intelligent model-based decision-making approaches. The proposed non-ideal heterogeneous mixing models capture continuity (heat and mass conservation), yet are extremely simple with few parameters, so they lend themselves to fast online prediction (with extrapolation capabilities) and regular recalibration. Further, they are more accurate than computational fluid dynamics (CFD) (60% less error) and symbolic regression data-driven models (73% less error). Real-time high-resolution sensor data are collected for observing spatiotemporal responses of state variables (conductivity, pH, and temperature) to transient influent shocks. Deterministic global dynamic optimization is used for training and recalibration of the non-ideal heterogeneous mixing models to guarantee the best-possible fits to the sensor data. The models are then deployed within standard model-predictive control and two economic model-predictive control strategies to demonstrate model-based decision-making for disturbance rejection and optimal operation of aeration in a continuous flow nitrification system utilizing high-resolution sensor data from several spatial positions. The new technology platform, consisting of high-resolution sensors, non-ideal heterogeneous mixing modeling, deterministic global dynamic optimization, and model-predictive control, offers superior performance over current approaches in water and wastewater treatment processes.

Deterministic global optimization with Gaussian processes embedded

Gaussian processes (Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the free variables and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the “MeLOn—MachineLearning Models for Optimization” toolbox (https://git.rwth-aachen.de/avt.svt/public/MeLOn).