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It is shown that quantum resonance can be used for combinatorial optimization. The advantage of the approach is in independence of the computing time upon the dimensionality of the problem. As an example, the solution to a constraint satisfaction problem of exponential complexity is demonstrated.
The Computer Aided Process Planning (CAPP) module was developed to facilitate and accelerate the process planning workflow for Automated Fiber Placement (AFP). CAPP assists process planners in identifying optimal starting point locations and layup strategies for each ply of a laminate. Ply optimization operates on measurement and scoring of geometry-based defects such as gaps, overlaps, angle deviation, and steering. This paper expands on the established framework for analyzing defect stack-up through thickness of a laminate. Four different combinatorial optimization algorithms are implemented and evaluated: (1) genetic algorithm, (2) differential evolution, (3) particle swarm, and (4) greedy search. The algorithms identify the optimal combination of ply-level layup strategies, by scoring potential laminates on defect stacking, using two different objective functions. A final optimization approach is also presented which trades some performance for a large gain in efficiency. These approaches are compared to a randomized combination using a complex tool surface in a virtual case study. The result is a streamlined methodology for comparing different laminate-level manufacturing strategies and minimizing the through thickness defect stack up.
Research supported by NASA Langley Research Center includes many applications of aerospace design optimization and is conducted by teams of applied mathematicians and aerospace engineers. This paper investigates the benefits from this combined expertise in solving combinatorial optimization problems. Applications range from the design of large space antennas to interior noise control. A typical problem, for example, seeks the optimal locations for vibration-damping devices on a large space structure and is expressed as a mixed/integer linear programming problem with more than 1500 design variables.
Research supported by NASA Langley Research Center includes many applications of aerospace design optimization and is conducted by teams of applied mathematicians and aerospace engineers. This paper investigates the benefits from this combined expertise in formulating and solving integer and combinatorial optimization problems. Applications range from the design of large space antennas to interior noise control. A typical problem. for example, seeks the optimal locations for vibration-damping devices on an orbiting platform and is expressed as a mixed/integer linear programming problem with more than 1500 design variables.
Research supported by NASA Langley Research Center includes many applications of aerospace design optimization and is conducted by teams of applied mathematicians and aerospace engineers. This paper investigates the benefits from this combined expertise in formulating and solving integer and combinatorial optimization problems. Applications range from the design of large space antennas to interior noise control. A typical problem, for example, seeks the optimal locations for vibration-damping devices on an orbiting platform and is expressed as a mixed/integer linear programming problem with more than 1500 design variables.
Libraries of liquid-phase catalyst precursor solutions were printed onto iridium-coated silicon substrates and evaluated for their effectiveness in catalyzing the growth of multi-walled carbon nanotubes (MWNTs) by chemical vapor deposition (CVD). The catalyst precursor solutions were composed of inorganic salts and a removable tri-block copolymer (EO)20(PO)70(EO)20 (EO = ethylene oxide, PO = propylene oxide) structure-directing agent (SDA), dissolved in ethanol/methanol mixtures. Sample libraries were quickly assayed using scanning electron microscopy after CVD growth to identify active catalysts and CVD conditions. Composition libraries and focus libraries were then constructed around the active spots identified in the discovery libraries to understand how catalyst precursor composition affects the yield, density, and quality of the nanotubes. Successful implementation of combinatorial optimization methods in the development of highly active, carbon nanotube catalysts is demonstrated, as well as the identification of catalyst formulations that lead to varying densities and shapes of aligned nanotube towers.
Recent Large Language Models (LLMs) have demonstrated impressive capabilities at tasks that require human intelligence and are a significant step towards human-like artificial intelligence (AI). Yet the performance of LLMs at reasoning tasks have been subpar and the reasoning capability of LLMs is a matter of significant debate. While it has been shown that the choice of the prompting technique to the LLM can alter its performance on a multitude of tasks, including reasoning, the best performing techniques require human-made prompts with the knowledge of the tasks at hand. We introduce a framework for what we call Combinatorial Reasoning (CR), a fully-automated prompting method, where reasons are sampled from an LLM pipeline and mapped into a Quadratic Unconstrained Binary Optimization (QUBO) problem. The framework investigates whether QUBO solutions can be profitably used to select a useful subset of the reasons to construct a Chain-of-Thought style prompt. We explore the acceleration of CR with specialized solvers. We also investigate the performance of simpler zero-shot strategies such as linear majority rule or random selection of reasons. Our preliminary study indicates that coupling a combinatorial solver to generative AI pipelines is an interesting avenue for AI reasoning and elucidates design principles for future CR methods.
The research proposed in this document investigated multiobjective optimization approaches based upon the Genetic Algorithm (GA). Several versions of the GA have been adopted for multiobjective design, but, prior to this research, there had not been significant comparisons of the most popular strategies. The research effort first generalized the two-branch tournament genetic algorithm in to an N-branch genetic algorithm, then the N-branch GA was compared with a version of the popular Multi-Objective Genetic Algorithm (MOGA). Because the genetic algorithm is well suited to combinatorial (mixed discrete / continuous) optimization problems, the GA can be used in the conceptual phase of design to combine selection (discrete variable) and sizing (continuous variable) tasks. Using a multiobjective formulation for the design of a 50-passenger aircraft to meet the competing objectives of minimizing takeoff gross weight and minimizing trip time, the GA generated a range of tradeoff designs that illustrate which aircraft features change from a low-weight, slow trip-time aircraft design to a heavy-weight, short trip-time aircraft design. Given the objective formulation and analysis methods used, the results of this study identify where turboprop-powered aircraft and turbofan-powered aircraft become more desirable for the 50 seat passenger application. This aircraft design application also begins to suggest how a combinatorial multiobjective optimization technique could be used to assist in the design of morphing aircraft.
Inaccuracies in the length of members and the diameters of joints of large truss reflector backup structures may produce unacceptable levels of surface distortion and member forces. However, if the member lengths and joint diameters can be measured accurately it is possible to configure the members and joints so that root-mean-square (rms) surface error and/or rms member forces is minimized. Following Greene and Haftka (1989) it is assumed that the force vector f is linearly proportional to the member length errors e(sub M) of dimension NMEMB (the number of members) and joint errors e(sub J) of dimension NJOINT (the number of joints), and that the best-fit displacement vector d is a linear function of f. Let NNODES denote the number of positions on the surface of the truss where error influences are measured. The solution of the problem is discussed. To classify, this problem was compared to a similar combinatorial optimization problem. In particular, when only the member length errors are considered, minimizing d(sup 2)(sub rms) is equivalent to the quadratic assignment problem. The quadratic assignment problem is a well known NP-complete problem in operations research literature. Hence minimizing d(sup 2)(sub rms) is is also an NP-complete problem. The focus of the research is the development of a simulated annealing algorithm to reduce d(sup 2)(sub rms). The plausibility of this technique is its recent success on a variety of NP-complete combinatorial optimization problems including the quadratic assignment problem. A physical analogy for simulated annealing is the way liquids freeze and crystallize. All computational experiments were done on a MicroVAX. The two interchange heuristic is very fast but produces widely varying results. The two and three interchange heuristic provides less variability in the final objective function values but runs much more slowly. Simulated annealing produced the best objective function values for every starting configuration and was faster than the two and three interchange heuristic.
In this paper we analyze the performance of the Quantum Adiabatic Evolution (QAE) algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, gamma = M / N. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (verses only energy) is used, and are able to show the existence of a dynamic threshold gamma = gammad, beyond which QAE should take an exponentially long time to find a solution. We compare the results for extended and simplified sets of landscapes and provide numerical evidence in support of our universality ansatz.
Combinatorial optimization is anticipated to be one of the primary use cases for quantum computation in the coming years. The Quantum Approximate Optimization Algorithm (QAOA) and Quantum Annealing (QA) have the potential to demonstrate significant run-time performance benefits over current state-of-the-art solutions. Using existing methods for characterizing classical optimization algorithms, we analyze solution quality obtained by solving Max-Cut problems using a quantum annealing device and gate-model quantum simulators and devices. This is used to guide the development of an advanced benchmarking framework for quantum computers designed to evaluate the trade-off between run-time execution performance and the solution quality for iterative hybrid quantum-classical applications. The framework generates performance profiles through effective visualizations that show performance progression as a function of time for various problem sizes and illustrates algorithm limitations uncovered by the benchmarking approach. The framework is an enhancement to the existing open-source QED-C Application-Oriented Benchmark suite and can connect to the open-source analysis libraries. The suite can be executed on various quantum simulators and quantum hardware systems.
Mapping the solution domain of n-finite elements into N-subdomains that may be processed in parallel by N-processors is an optimal one if the subdomain decomposition results in a well-balanced workload distribution among the processors. The problem is discussed in the context of irregular finite element domains as an important aspect of the efficient utilization of the capabilities of emerging multiprocessor computers. Finding the optimal mapping is an intractable combinatorial optimization problem, for which a satisfactory approximate solution is obtained here by analogy to a method used in statistical mechanics for simulating the annealing process in solids. The simulated annealing analogy and algorithm are described, and numerical results are given for mapping an irregular two-dimensional finite element domain containing a singularity onto the Hypercube computer.
Carbon nanotube (CNT) exhibits extraordinary mechanical and unique electronic properties and offers significant potential for structural, sensor, and nanoelectronics applications. An overview of CNT, growth methods, properties and applications is provided. Single-wall, and multi-wall CNTs have been grown by chemical vapor deposition. Catalyst development and optimization has been accomplished using combinatorial optimization methods. CNT has also been grown from the tips of silicon cantilevers for use in atomic force microscopy.
Farhi et al. recently proposed a class of quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), for approximately solving combinatorial optimization problems. A level-p QAOA circuit consists of steps in which a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The 2p times for which these two Hamiltonians are applied are the parameters of the algorithm. As p increases, however, the parameter search space grows quickly. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here, we analytically and numerically study parameter setting for QAOA applied to MAXCUT. For level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MAXCUT, the Ring of Disagrees, or the 1D antiferromagnetic ring, we provide an analysis for arbitrarily high level. Using a Fermionic representation, the evolution of the system under QAOA translates into quantum optimal control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of QAOA for any p. It also greatly simplifies numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional sub-manifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.
Researchers at NASA Langley Research Center have extensive experience using active structural acoustic control (ASAC) for aircraft interior noise reduction. One aspect of ASAC involves the selection of optimum locations for microphone sensors and force actuators. This paper explains the importance of sensor/actuator selection, reviews optimization techniques, and summarizes experimental and numerical results. Three combinatorial optimization problems are described. Two involve the determination of the number and position of piezoelectric actuators, and the other involves the determination of the number and location of the sensors. For each case, a solution method is suggested, and typical results are examined. The first case, a simplified problem with simulated data, is used to illustrate the method. The second and third cases are more representative of the potential of the method and use measured data. The three case studies and laboratory test results establish the usefulness of the numerical methods.
A combinatorial optimization methodology is developed, which enables the efficient use of hypercube multiprocessors onboard mobile intelligent robots dedicated to time-critical missions. The methodology is implemented in terms of large-scale concurrent algorithms based either on fast simulated annealing, or on nonlinear asynchronous neural networks. In particular, analytic expressions are given for the effect of single-neuron perturbations on the systems' configuration energy. Compact neuromorphic data structures are used to model effects such as precedence constraints, processor idling times, and task-schedule overlaps. Results for a typical robot-dynamics benchmark are presented.
The optimal placement of discrete actuators and sensors is posed as a combinatorial optimization problem. Two examples for truss structures were used for illustration; the first dealt with the optimal placement of passive dampers along existing truss members, and the second dealt with the optimal placement of a combination of a set of actuators and a set of sensors. Except for the simplest problems, an exact solution by enumeration involves a very large number of function evaluations, and is therefore computationally intractable. By contrast, the simulated annealing heuristic involves far fewer evaluations and is best suited for the class of problems considered. As an optimization tool, the effectiveness of the algorithm is enhanced by introducing a number of rules that incorporate knowledge about the physical behavior of the problem. Some of the suggested rules are necessarily problem dependent.
Distortions in truss structures can result from random errors in element lengths that are typical of a manufacturing process. These distortions may be minimized by an optimal selection of elements from those available for placement between the prescribed nodes - a combinatorial optimization problem requiring significant investment of computational resource for all but the smallest problems. The present paper describes a formulation in which near-optimal element assignments are obtained as minimum-energy stable states, of an analogous Hopfield neural network. This requires mapping of the optimization problem into an energy function of the appropriate Liapunov form. The computational architecture is ideally suited to a parallel processor implementation and offers significant savings in computational effort. A numerical implementation of the approach is discussed with reference to planar truss problems.