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A combinatorial optimization problem is trying to find out the value (combination) of variables that optimizes an index (value) from among many options under various constraints.

What is optimization? In the mathematics for annealing machines, I will explain the necessary mathematics to use an annealing machine based on the answer to that question.

The optimization process using this annealing machine involves several instances where mathematics is utilized.

I will explain the process flow and mathematical background of optimization using annealing machines and other optimization methods.

I will explain the first step of the mathematics necessary for optimization, along with Python programs that illustrate algebraic notation.

Chapter 2 focuses on understanding the mathematical knowledge necessary for grasping the objective function. First, it is about preparing your mind to accept without resistance what the equations intend to convey.

Most of the information devices we use today, such as PCs and smartphones, web servers that send out information on websites, IoT devices, and mission-critical systems that control power supply, are processed by computers called Neumann-type computers.

The Ising model is the simplest model in a field of physics 'statistical mechanics'.

With the outbreak of COVID-19, we used an annealing machine to create shifts for our researchers under constraints that took into account the newly created risks of infection.

Reinsurance is critical against natural disaster risks that have increased in recent years. We have used an annealing machine to formulate a reinsurance portfolio utilizing the vast amount of data collected by insurance companies.

I will explain the mathematics for creating an objective function, understanding it, the mathematical knowledge required to convert it into the Ising model, and the mathematical knowledge related to optimization methods.

Let me explain how to formulate the equation in light of the case of the 'Image Noise Reduction' demo application on this site.

Not every optimization problem can be dramatically improved with an annealing machine.

A linear programming problem is one that can be represented by a linear function.

The aforementioned first-order equations can be easily solved by other methods, whereas the second-order optimization problem is considered to be efficiently solved by an annealing machine.

So when does this second-order term appear in a real problem?

What is a typical combinatorial optimization problem? Many people may answer with the 'traveling salesman problem'.

Both machine learning, which is often used in AI, and annealing machines, which perform combinatorial optimization, are computers that perform difficult processing on behalf of humans.

An annealing machine is a technique for solving combinatorial optimization problems through a mechanism in which the ground state of the Ising model (the state with the lowest energy) gives the optimal solution by setting preconditions that result in the optimal solution.

In the second part, we will discuss the structural specifications and technical challenges of the annealing machine that actually derives the ground state of the Ising model.

When applying an annealing machine solution of an optimization problem to a real-world problem, the ultimate goal is to improve the user's work through the optimization process.

In this section, we will look back at the roles that hardware and software have played in the shift from the dawn of annealing machines to their widespread use, and also consider the roles they will play in the future.

This column introduces recommended books outside of Annealing Cloud Web that you should deepen your knowledge in order to actually use an annealing machine.

In general, computers often employ procedures and algorithms that lead to the only correct solution (exact solution), but let's learn about the different heuristic features and the significance of annealing machines in this perspective.

Did you know that annealing machines are already being used in business, not just for verification, but as a means to solve real-world problems? This article will cover examples of applications that have been published so far.

In this column, I would like to introduce the know-how for applying mathematical optimization technology, including CMOS annealing, to issues that require human decision-making.

This column is filled with hints for advancing issue clarification and requirement definition, so if you're unsure where to start or how to articulate your thoughts as you progress in your study of mathematical optimization, please take a moment to read it.

In order to encourage more people to use an annealing machine, we have published an article explaining the process of solving optimization problems with annealing machines for beginners.

In this article, we explain how to organize information of the problem that you want to solve, which will be done in the requirements definition stage, using the 'school class timetable' as a motif.

Solving a combinatorial optimization problem with an annealing machine requires a process called Ising model formulation.

In this article, we will create input data for the number partitioning problem that we have made 'formulation' in the previous article.

Let's input the data for the number partitioning problem created in the 'Input Data Preparation' section into the CMOS annealing machine to solve it.

This tutorial explains the processing of the annealing machine in the demonstration application 'Ising editor'

This tutorial explains the processing of the annealing machine in the demonstration application 'Image noise reduction'.

This tutorial explains the processing of the annealing machine in the demonstration application 'Building network robustness'

Annealing Cloud Web API provides annealing machine operations using the Ising model as a Web API that communicates via HTTP.

In Ising editor, you can set parameters and schedule of CMOS annealing machine and perform annealing.

The Ising editor is a learning tool that provides a visual representation of the CMOS annealing machine in action, but it is a unique tool for visually seeing what to input to the CMOS annealing machine and how to obtain a solution.

This demo app is to solve the problem of estimating the original image from a noisy black and white image with an annealing machine.

This demo app is to solve combinatorial optimization problems with an annealing machine taking as an example a problem called vertex cover problem of graph.

This article provides an exercise to realize a signal control system to relieve traffic congestion as a signal control system by processing it with a CMOS annealing machine as a combinatorial optimization problem.

With the Fitness diagnostic tool, you can confirm whether the optimization examples or problems you are familiar with are actually compatible with an annealing machine.

One of the use cases for an annealing machine is "optimizing researcher shifts while considering COVID-19 infection control." Here, I will explain how to use a decision architecture design framework in practice.

This will provide an overview of technologies that substitute human decision-making, similar to annealing machines, and explain the characteristics and advantages of annealing machines among them.

This explanation will cover how to use the Fixstars Amplify SDK to run a CMOS annealing machine, focusing on the image noise reduction problem. It will also provide tips for achieving performance using king's graphs.