Optimisation Problems Research Articles

Sample-Constrained Black Box Optimization for Audio Personalization

We consider the problem of personalizing audio to maximize user experience. Briefly, we aim to find a filter h*, which applied to any music or speech, will maximize the user’s satisfaction. This is a black-box optimization problem since the user’s satisfaction function is unknown. Substantive work has been done on this topic where the key idea is to play audio samples to the user, each shaped by a different filter hi, and query the user for their satisfaction scores f(hi). A family of “surrogate” functions is then designed to fit these scores and the optimization method gradually refines these functions to arrive at the filter ˆh* that maximizes satisfaction. In certain applications, we observe that a second type of querying is possible where users can tell us the individual elements h*[j] of the optimal filter h*. Consider an analogy from cooking where the goal is to cook a recipe that maximizes user satisfaction. A user can be asked to score various cooked recipes (e.g., tofu fried rice) or to score individual ingredients (say, salt, sugar, rice, chicken, etc.). Given a budget of B queries, where a query can be of either type, our goal is to find the recipe that will maximize this user’s satisfaction. Our proposal builds on Sparse Gaussian Process Regression (GPR) and shows how a hybrid approach can outperform any one type of querying. Our results are validated through simulations and real world experiments, where volunteers gave feedback on music/speech audio and were able to achieve high satisfaction levels. We believe this idea of hybrid querying opens new problems in black-box optimization and solutions can benefit other applications beyond audio personalization.

Mixed-integer optimisation of graph neural networks for computer-aided molecular design

ReLU neural networks have been modelled as constraints in mixed integer linear programming (MILP), enabling surrogate-based optimisation in various domains and efficient solution of machine learning certification problems. However, previous works are mostly limited to MLPs. Graph neural networks (GNNs) can learn from non-euclidean data structures such as molecular structures efficiently and are thus highly relevant to computer-aided molecular design (CAMD). We propose a bilinear formulation for ReLU Graph Convolutional Neural Networks and a MILP formulation for ReLU GraphSAGE models. These formulations enable solving optimisation problems with trained GNNs embedded to global optimality. We apply our optimisation approach to an illustrative CAMD case study where the formulations of the trained GNNs are used to design molecules with optimal boiling points.

Multi-objective optimization of machining parameters based on an improved Hopfield neural network for STEP-NC manufacturing

Aiming at solving the existing problems of machining parameters optimization for STEP-NC manufacturing, a method for multi-objective optimization of machining parameters based on an improved Hopfield neural network (IHNN) for STEP-NC manufacturing is proposed. In this method, a multi-objective optimization mathematical model of machining parameters compliant with STEP-NC is firstly established taking machining energy, machining time and machining cost as optimization objectives. Next, the IHNN for multi-objective optimization of STEP-NC machining parameters combining with Pareto theory, improved immune algorithm and non-monotone activation function is designed. Based on it, the optimal Pareto solutions of STEP-NC machining parameters are obtained, which intelligently realizes the multi-objective optimization of STEP-NC machining parameters and provides a decision support for the decision-maker. Finally, performance comparisons among the IHNN, classic non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective particle swarm optimization algorithm (MOPSO) are done by classic test functions with three objectives and multiple constraints which correspond to the mathematical model established in this paper, and its effectiveness and feasibility is verified by case study.

Physics-informed shape optimization using coordinate projection

The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework’s capability as a viable tool for shape optimization in complex material design tasks.

Decoupled magnetic control of spherical tokamak divertors via vacuum harmonic constraints

Power exhaust is a critical challenge for spherical tokamak reactors, making the design, optimisation and control of advanced divertor configurations crucial. These tasks are greatly simplified if the poloidal magnetic fields in the core and divertor regions can be varied independently. We present a novel method which facilitates decoupling of the core plasma equilibrium from the divertor geometry optimisation and control, using vacuum spherical harmonic (SH) constraints. This has the advantage that it avoids iterative solution of the Grad–Shafranov equation, making it easy to use, rapid and reliable. By comparing a large number of MAST-U equilibrium reconstructions against their approximations using SHs, a small number ( ∼4 ) of harmonics is found to be sufficient to closely reproduce the plasma boundary shape. We show experimentally that poloidal field changes designed to leave harmonics unaffected indeed have no effect on the core plasma shape. When augmented with divertor geometry constraints, this approach gives a powerful tool for creating advanced magnetic configurations, and its simplicity brings improvements in speed and robustness when solving coil position optimisation problems. We discuss the clear benefits to real-time feedback control, feed-forward scenario design and coilset optimisation with a view to future reactors.

Structural and morphological organization of dry valley of the Cretaceous south of the Central Chernozem Region

The peculiarities of origin, morphological and genetic varieties and structural and dynamic interrelations of dry valley landscapes of the Cretaceous south of the Central Chernozem Region are considered. Modern ideas about genesis, morphology, development and landscape structure of sukhodols are presented. Paradynamic interrelations of structural elements of basin landscape systems are emphasized. The necessity of taking into account landscape specificity of dry lands when solving the problems of nature management and optimization of landscape-ecological situation in the Cretaceous south of the Central Chernozem Region is emphasized. As an example, we analyzed the component, territorial and altitudinal organization of natural complexes of a large sukhodols of the Kuvshin dry valley located in the Podgorensky district of the Voronezh Region. The main factors of landscape differentiation are established, among which are peculiarities of morpholithogenic base, character of biostrom development, modern economic use. The problems of identification of local landscape boundaries and approaches to their solution have been noted. Based on the results of the research, 7 families and 45 species of landtypes, as well as altitudinal landscape layers and sub-layers were identified within the valley. Recommendations on changing the conservation status of the territory are given.

An in-principle super-polynomial quantum advantage for approximating combinatorial optimization problems via computational learning theory.

It is unclear to what extent quantum algorithms can outperform classical algorithms for problems of combinatorial optimization. In this work, by resorting to computational learning theory and cryptographic notions, we give a fully constructive proof that quantum computers feature a super-polynomial advantage over classical computers in approximating combinatorial optimization problems. Specifically, by building on seminal work by Kearns and Valiant, we provide special instances that are hard for classical computers to approximate up to polynomial factors. Simultaneously, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The quantum advantage in this work is ultimately borrowed from Shor's quantum algorithm for factoring. We introduce an explicit and comprehensive end-to-end construction for the advantage bearing instances. For these instances, quantum computers have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms.

Benchmarking problems for robust discrete optimization

Robust discrete optimization is a highly active field of research where a plenitude of combinations between decision criteria, uncertainty sets and underlying nominal problems are considered. Usually, a robust problem becomes harder to solve than its nominal counterpart, even if it remains in the same complexity class. For this reason, specialized solution algorithms have been developed. To further drive the development of stronger solution algorithms and to facilitate the comparison between methods, a set of benchmark instances is necessary but so far missing. In this paper we propose a further step towards this goal by proposing several instance generation procedures for combinations of min–max, min–max regret, two-stage and recoverable robustness with interval, discrete, budgeted or ellipsoidal uncertainty sets. Besides sampling methods that go beyond the simple uniform sampling method that is the de-facto standard to produce instances, also optimization models to construct hard instances are considered. Using a selection problem for the nominal ground problem, we are able to generate instances that are several orders of magnitudes harder to solve than uniformly sampled instances when solving them with a general mixed-integer programming solver. All instances and generator codes are made available online.

A comparative study of maximum matching algorithms

One of the central problems of discrete optimization is the problem of determining the maximal matching, which can be solved in various ways, depending on the structure of the initial data. In this article, a comparative study of the efficiency of maximal flow algorithms (Ford-Fulkerson) and the direct algorithm for determining maximal matching (Even & Kariv) on unweighted bipartite graphs is conducted. The analysis is based on a set of high-difficulty computing competition problems. Therefore, the results will be useful not only to those interested in the maximal matching problem but also to all those who are preparing to participate in various programming competitions.

A comprehensive high‐level automated driving assistance system with integrated multi‐functionality

AbstractAdvanced Driver Assistance Systems (ADAS) have gained substantial attention in recent years. However, the integration mechanism of multiple functions within ADAS remains unexplored, and the full potential of its functionality remains underutilised. This paper presents a novel multi‐functional integrated High‐level Automated Driving Assistance System that combines the Cruise Control (CC), Adaptive Cruise Control (ACC), Automated Emergency Brake (AEB), and Automated Lane Change (ALC) functions. The presented system utilises a hierarchical framework. The extension multi‐mode switch strategy is established as the superior module and the Event‐Triggered Model Predictive Controller (ETMPC) is designed as the inferior controller. The CC, ACC, and ALC functions are effectively utilised to enhance traffic efficiency, while the AEB function ensures driving safety. To address the time constraints of conventional Model Predictive Control, an event‐trigger mechanism is proposed to reduce computational load. Simulations are conducted using the CarSim and Matlab platforms. The study results demonstrate significant improvements in both safety and traffic efficiency compared to conventional ADAS strategies. Furthermore, the proposed ETMPC method significantly reduces the frequency of solving Optimisation Problems and decreases online computation costs.

Connectivity conservation planning through deep reinforcement learning

Abstract The United Nations has declared 2021–2030 the decade on ecosystem restoration with the aim of preventing, stopping and reversing the degradation of the ecosystems of the world, often caused by the fragmentation of natural landscapes. Human activities separate and surround habitats, making them too small to sustain viable animal populations or too far apart to enable foraging and gene flow. Despite the need for strategies to solve fragmentation, it remains unclear how to efficiently reconnect nature. In this paper, we illustrate the potential of deep reinforcement learning (DRL) to tackle the spatial optimisation aspect of connectivity conservation planning. The propensity of spatial optimisation problems to explode in complexity depending on the number of input variables and their states is and will continue to be one of its most serious obstacles. DRL is an emerging class of methods focused on training deep neural networks to solve decision‐making tasks and has been used to learn good heuristics for complex optimisation problems. While the potential of DRL to optimise conservation decisions seems huge, only few examples of its application exist. We applied DRL to two real‐world raster datasets in a connectivity planning setting, targeting graph‐based connectivity indices for optimisation. We show that DRL converges to the known optimums in a small example where the objective is the overall improvement of the Integral Index of Connectivity and the only constraint is the budget. We also show that DRL approximates high‐quality solutions on a large example with additional cost and spatial configuration constraints where the more complex Probability of Connectivity Index is targeted. To the best of our knowledge, there is no software that can target this index for optimisation on raster data of this size. DRL can be used to approximate good solutions in complex spatial optimisation problems even when the conservation feature is non‐linear like graph‐based indices. Furthermore, our methodology decouples the optimisation process and the index calculation, so it can potentially target any other conservation feature implemented in current or future software.

Advancements in Unsupervised Learning: Mode-Assisted Quantum Restricted Boltzmann Machines Leveraging Neuromorphic Computing on the Dynex Platform

The integration of neuromorphic computing into the Dynex platform signifies a transformative step in computational technology, particularly in the realms of machine learning and optimization. This advanced platform leverages the unique attributes of neuromorphic dynamics, utilizing neuromorphic annealing - a technique divergent from conventional computing methods - to adeptly address intricate problems in discrete optimization, sampling, and machine learning. Our research concentrates on enhancing the training process of Restricted Boltzmann Machines (RBMs), a category of generative models traditionally challenged by the intricacy of computing their gradient. Our proposed methodology, termed “quantum mode training”, blends standard gradient updates with an off-gradient direction derived from RBM ground state samples. This approach significantly improves the training efficacy of RBMs, outperforming traditional gradient methods in terms of speed, stability, and minimized converged relative entropy (KL divergence). This study not only highlights the capabilities of the Dynex platform in progressing unsupervised learning techniques but also contributes substantially to the broader comprehension and utilization of neuromorphic computing in complex computational tasks.

Quantum annealing-driven branch and bound for the single machine total weighted number of tardy jobs scheduling problem

In the paper we present a new approach to solving NP-hard problems of discrete optimization adapted to the architecture of quantum processors (QPU, Quantum Processor Unit) implementing hardware quantum annealing. This approach is based on the use of the quantum annealing metaheuristic in the exact branch and bound algorithm to compute the lower and upper bounds of the objective function. To determine the lower bound, a new method of defining the Lagrange function for the dual problem (the generalized discrete knapsack problem) was used, the value of which is calculated on the QPU of a quantum machine. In turn, to determine the upper bound, we formulate an appropriate task in the form of binary quadratic programming with constraints.Despite the fact that the results generated by the quantum machine are probabilistic, the hybrid method of algorithm construction proposed in the paper, using alternately a CPU and QPU, guarantees the optimal solution. As a case study we consider the NP-hard single machine scheduling problem with minimizing the weighted number of tardy jobs. The performed computational experiments showed that optimal solutions were already obtained in the root of the solution tree, and the values of the lower and upper bounds differ by only a few percent.

Benchmark problems for large-scale constrained multi-objective optimization with baseline results

The interests in evolutionary constrained multiobjective optimization are rapidly increasing during the past two decades. However, most related studies are limited to small-scale problems, despite the fact that many practical problems contain large-scale decision variables. Although several large-scale constrained multi-objective evolutionary algorithms (CMOEAs) have been developed, they are still tested on benchmarks that are designed for small-scale problems without the features of large-scale problems. To promote the research on large-scale constrained multi-objective optimization (LSCMO), this paper proposes a new LSCMO benchmark based on the design principles of large-scale multi-objective optimization and constrained multi-objective optimization. In this benchmark, more realistic features are considered, such as mixed linkages between constraint variables and unconstrained variables, imbalanced contributions of variables to the objectives, varying number constraint functions. Besides, to better solve the proposed benchmark, a bidirectional sampling strategy is proposed, where a convergence direction sampling and a diversity direction sampling are used to accelerate the convergence and maintain diversity respectively. Furthermore, the proposed bidirectional sampling strategy is embedded into an existing CMOEA to improve the search ability of algorithm in the large-scale search space with constraints. In experiments, the proposed algorithm is compared with several latest peer algorithms, and the results verify that the designed benchmark functions can effectively test the performance of algorithms and the proposed algorithm can better tackle the new benchmark. Finally, the proposed algorithm is used to solve the network structure control-based personalized drug target recognition problems with more than 2000 decision variables, and results show its superiority.

Continuous-time quantum walks for MAX-CUT are hot

By exploiting the link between time-independent Hamiltonians and thermalisation, heuristic predictions on the performance of continuous-time quantum walks for MAX-CUT are made. The resulting predictions depend on the number of triangles in the underlying MAX-CUT graph. We extend these results to the time-dependent setting with multi-stage quantum walks and Floquet systems. The approach followed here provides a novel way of understanding the role of unitary dynamics in tackling combinatorial optimisation problems with continuous-time quantum algorithms.

An expressive ansatz for low-depth quantum approximate optimisation

The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.

Rapid quantum approaches for combinatorial optimisation inspired by optimal state-transfer

We propose a new design heuristic to tackle combinatorial optimisation problems, inspired by Hamiltonians for optimal state-transfer. The result is a rapid approximate optimisation algorithm. We provide numerical evidence of the success of this new design heuristic. We find this approach results in a better approximation ratio than the Quantum Approximate Optimisation Algorithm at lowest depth for the majority of problem instances considered, while utilising comparable resources. This opens the door to investigating new approaches for tackling combinatorial optimisation problems, distinct from adiabatic-influenced approaches.

Multi-objective disassembly line design and optimisation considering energy efficiency and human factors

Green manufacturing and remanufacturing revolve around recycling end-of-life (EOL) products. One crucial aspect of this process is the disassembly line balancing problem (DLBP), which involves assigning disassembly tasks to workstations in a logical order. However, current DLBP models often overlook operator-related factors and energy efficiency. This paper introduces a multi-objective DLBP model that considers both operator and energy factors. The model aims to minimise disassembly line idle rate, improve line smoothness, reduce energy consumption, and balance operator workload under multiple constraints. Since this problem is NP-hard, we also develop a hybrid intelligent algorithm that combines simulated annealing and the water cycle algorithm (SWCA) to solve DLBP effectively using real-world case disassembly. For multi-objective optimisation problems, choosing multiple non-inferior solutions can be challenging. In this work, we embrace an ‘optimisation before decision-making' approach, integrating multi-criteria decision-making techniques with neural networks optimised by a particle swarm optimisation (PSO) algorithm to identify the most satisfactory solution.

Validation and characterization of algorithms and software for photonics inverse design

In this work, we present a reproducible suite of test problems for large-scale optimization (“inverse design” and “topology optimization”) in photonics, where the prevalence of irregular, non-intuitive geometries can otherwise make it challenging to be confident that new algorithms and software are functioning as claimed. We include test problems that exercise a wide array of physical and mathematical features—far-field metalenses, 2d and 3d mode converters, resonant emission and focusing, and dispersion/eigenvalue engineering—and introduce an a posteriori lengthscale metric for comparing designs produced by disparate algorithms. For each problem, we incorporate cross-checks against multiple independent software packages and algorithms, and reproducible designs and their validations scripts are included. We believe that this suite should make it much easier to develop, validate, and gain trust in future inverse-design approaches and software.

Selected Problems of Optimization of a Switched Reluctance Motor for an Electric Vehicle using Analytical Calculations

A model and an example of optimization of a 6/4 type SRM, designed for an electric vehicle, is described in the paper. The modelling procedure, based on simplified analytical calculations, allows for estimation of the torque, efficiency and acoustic noise of the motor, taking into account the magnetic non-linearity. Some problems arising from the existence of mutual inductances are described as well. Calculations have been performed in MATLAB environment, and verified by means of FEM analysis.