Optimization Algorithms And Applications

[Argelia Long]

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Hybrid classical quantum optimization methods have become an important tool for efficiently solving problems in the current generation of noisy intermediate-scale quantum computers. These methods use an optimization algorithm executed in a classical computer, fed with values of the objective function obtained in a quantum processor. A proper choice of optimization algorithm is essential to achieve good performance. Here, we review the use of first-order, second-order, and quantum natural gradient stochastic optimization methods, which are defined in the field of real numbers, and propose stochastic algorithms defined in the field of complex numbers. The performance of all methods is evaluated by means of their application to variational quantum eigensolver, quantum control of quantum states, and quantum state estimation. In general, complex number optimization algorithms perform best, with first-order complex algorithms consistently achieving the best performance, closely followed by complex quantum natural algorithms, which do not require expensive hyperparameter calibration. In particular, the scalar formulation of the complex quantum natural algorithm allows to achieve good performance with low classical computational cost.

The mean (top row) and median (bottom row) of the energy (in arbitrary units) as a function of the number of iterations obtained through the VQE for the Heisenberg Hamiltonian in a ten-qubit ring configuration using vanilla optimization algorithms. The shaded areas represent the variance (top row) and the interquartile range (bottom row). The dashed line indicates the exact minimum. The statistics are obtained from a sample of 102 randomly generated states to estimate the minimum energy. The measurements in each circuit were estimated with 2104 shots. The values of the gain coefficients and postprocessing class can be found in Table 1.

The mean (top row) and median (bottom row) of the energy as a function of the number of iterations obtained through the VQE for the Heisenberg Hamiltonian in a ten-qubit ring configuration using improved optimization algorithms. The shaded areas represent the variance (top row) and the interquartile range (bottom row). The dashed line indicates the exact minimum. The statistics are obtained from a sample of 102 randomly generated states to estimate the minimum energy. The measurements in each circuit were estimated with 2104 shots. The values of the gain coefficients, postprocessing class, and the setting of resampling and blocking can be found in Table 2.

The mean (top row) and median (bottom row) of the infidelity as a function of the number of iterations obtained by using SGQT to estimate six-qubit states and vanilla optimization algorithms. Shaded areas represent variance (top row) and interquartile range (bottom row). Statistical indicators are calculated from a sample of 102 Haar-uniform distributed pairs of unknown and initial guess states. Measurements of the infidelity are simulated with a binomial distribution with N=2104 shots. The values of the gain coefficients and postprocessing class can be found in Table 5.

The mean (top row) and median (bottom row) of the infidelity as a function of the number of iterations obtained by using SGQT to estimate six-qubit states and improved optimization algorithms. Shaded areas represent variance (top row) and interquartile range (bottom row). Statistical indicators are calculated from a sample of 102 Haar-uniform distributed pairs of unknown and initial guess states. Measurements of the infidelity are simulated with a binomial distribution with N=2104 shots. The values of the gain coefficients, postprocessing class, and the setting of resampling and blocking can be found in Table 6.

The objective of this course is to learn to recognize, transform and solve a broad class of convex optimization problems arising in various fields such as machine learning, finance or signal processing. The course starts with a basic primer on convex analysis followed by a quick overview of convex duality theory. The second half of the course is focused on algorithms, including first-order and interior point methods, together with bounds on their complexity.The course ends with illustrations of these techniques in various applications.

DM3: Due Monday November 20. Homework 3. Please submit your code on Gradescope as a PDF. For Jupyter notebooks, this can be done by downloading the notebook as a PDF. Otherwise a screenshot of your code is acceptable. Please ensure the theoretical results, plots and comments on numerical results are easily identifiable and clearly readable. For numerical exercises, you can use either Julia, Python or MATLAB.

Optimization algorithms are mathematical procedures used to find optimal solutions to complex problems. These algorithms are integral to many areas of science and industry, including artificial intelligence, machine learning, operations research, and data science. They are frequently used to improve systems and processes by minimizing or maximizing some function.

Optimization algorithms work by iteratively improving upon a candidate solution until a predetermined optimum level is reached, or resources are depleted. The optimization can be directed at reducing costs or time taken, increasing efficiency or profit, or maximizing resource use. Some commonly used optimization algorithms include linear programming, convex optimization, integer programming, and stochastic optimization.

Optimization algorithms can have a transformative impact on businesses. They can streamline operations, improve decision-making, and boost profitability by finding the most efficient ways to allocate resources. Their applications are vast and can span from optimizing supply chains and production processes to enhancing customer satisfaction and even guiding strategic planning.

Despite their advantages, optimization algorithms do have limitations. Some problems may be too complex and can't be solved to optimality. Other challenges include the requirement for significant computational power and the need for careful modeling to ensure that the algorithm accurately represents the real-world problem.

In the context of a data lakehouse, optimization algorithms can provide significant value. They can optimize data processing and analytics tasks, ensuring that resources are used efficiently and performance is maximized. For example, query optimization algorithms can streamline SQL queries to make data retrieval more efficient. Additionally, optimization algorithms can aid in workload management, helping to balance computational load and ensure smooth operation of the data lakehouse.

By optimizing the use of resources, optimization algorithms can greatly enhance system performance. They allow systems to do more with less, whether that is processing larger volumes of data, performing more complex computations or running tasks more quickly.

Abstract: Particle swarm optimization (PSO) is a popular heuristic method, which is capable of effectively dealing with various optimization problems. A detailed overview of the original PSO and some PSO variant algorithms is presented in this paper. An up-to-date review is provided on the development of PSO variants, which include four types i.e., the adjustment of control parameters, the newly-designed updating strategies, the topological structures, and the hybridization with other optimization algorithms. A general overview of some selected applications (e.g., robotics, energy systems, power systems, and data analytics) of the PSO algorithms is also given. In this paper, some possible future research topics of the PSO algorithms are also introduced.

Optimization plays a critical role in a variety of research fields such as mathematics, economics, engineering, and computer science. Recognizing as a popular class of optimization techniques, the evolutionary computation (EC) methods have behaved competitive performance in effectively tackling optimization problems in an easy way. So far, the EC methods have been widely applied in numerous research fields thanks to their strong abilities in finding the optimal solutions [1]. Among the EC algorithms, some algorithms which are based on biological behaviours (e.g., the genetic algorithm (GA) [2], the ant colony optimization (ACO) algorithm [3] and the particle swarm optimization (PSO) algorithm [4,5]) have been well-adopted in a number of research areas e.g., energy systems, robotics, aerospace engineering and artificial intelligence.

As a population-based EC method, PSO is developed on the basis of the mimics of social behaviours e.g., the birds-flocking phenomenon and the fish-schooling phenomenon. Notably, the potential optimization solution is represented by a particle (also called as individual). During the searching process, each individual learns from the "movement experience" of itself and others. It should be mentioned that the advantages of PSO can be summarised into three aspects: (1) the number of parameters required to be adjusted is relatively few; (2) the convergence rate of the PSO algorithm is relatively fast; and (3) the implementation of the PSO algorithm is simple [6,7]. Owing to its technical merits and easy implementation, PSO has become a widely-used technique for tackling optimization problems in recent years [8-10].

Unfortunately, many population-based EC algorithms face the challenging problem that the potential solutions being easily trapped into the local optima especially under complex and high-dimensional scenarios. As a well-known EC algorithm, PSO is not an exception. As a result, developing new PSO methods has become a seemingly reasonable way to deal with the premature convergence puzzle [11-16]. For example, a group of PSO variants have been put forward by modifying the parameters [13-15]. In [14, 15], a linear decreasing mechanism has been proposed to alter the inertia weight, leading to a proper balance between the global discovery and the local detection. In [13], a novel optimizer has been introduced by presenting a time-varying strategy to adjust the acceleration coefficients, which enhances the global search and convergence performance.

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