Table of contents
2. Competition time: from 18:00 on September 7, 2023 to 20:00 on September 10, 2023
See this business card for detailed ideas, and update as soon as the game starts
1. Question C thinking model: After the game starts, update it as soon as possible, and get the business card at the end of the article
2. Competition time: from 18:00 on September 7, 2023 to 20:00 on September 10, 2023
3 Common Mathematical and Modeling Problems in National Undergraduate Mathematical Contest in Modeling
classification model
optimization model
predictive model
evaluation model
3.1 Classification problem
Discriminant analysis:
Also known as the "discrimination method", it is a multivariate statistical analysis method to distinguish the type of a research object according to various eigenvalues of a certain research object under the condition of a certain classification.
The basic principle is to establish one or more discriminant functions according to certain discriminant criteria; use a large amount of data of the research object to determine the undetermined coefficients in the discriminant function, and calculate the discriminant index; according to this, it can be determined which category a certain sample belongs to. When a new sample data is obtained, it is necessary to determine which of the known types the sample belongs to. This type of problem belongs to the discriminant analysis problem.
Cluster analysis:
Clustering analysis or clustering is to divide similar objects into different groups or more subsets through static classification, so that the member objects in the same subset have similar attributes, which are commonly included in the coordinate system Shorter spatial distances etc.
Cluster analysis itself is not a specific algorithm, but a general task to be solved. It can be achieved with different algorithms that vary greatly in terms of understanding what constitutes clusters and how to find them efficiently.
Neural Network Classification:
BP neural network is a neural network learning algorithm. It is a hierarchical neural network composed of an input layer, an intermediate layer, and an output layer, and the intermediate layer can be extended to multiple layers. RBF (Radial Basis Function) neural network: The radial basis function (RBF-Radial Basis Function) neural network is a three-layer feed-forward network with a single hidden layer. It simulates the neural network structure in the human brain with locally adjusted, mutually overlapping receptive fields. Perceptron neural network: It is a neural network with a single layer of computational neurons, and the transfer function of the network is a linear threshold unit. It is mainly used to simulate the perceptual characteristics of the human brain. Linear neural network: It is a relatively simple neural network consisting of one or more linear neurons. A linear function is used as the transfer function, so the output can be any value. Self-organizing neural network: self-organizing neural network includes self-organizing competition network, self-organizing feature map network, learning vector quantization and other network structure forms. K-nearest neighbor algorithm: K-nearest neighbor classification algorithm is a relatively mature method in theory and one of the simplest machine learning algorithms.
3.2 Optimization problem
Linear programming:
Mathematical theories and methods for studying the extremum problems of linear objective functions under linear constraints. English abbreviation LP. It is an important branch of operations research, widely used in military operations, economic analysis, management and engineering technology. Modeling method: List the constraints and objective function; draw the feasible region represented by the constraints; find the optimal solution and optimal value of the objective function in the feasible region.
Nonlinear programming:
Nonlinear programming is a mathematical programming with nonlinear constraints or objective functions, and is an important branch of operations research. Nonlinear programming studies the extremum problem of an n-ary real function under a set of constraints of equality or inequality, and at least one of the objective function and the constraints is a nonlinear function of unknown quantity. The case where both the objective function and the constraints are linear functions is called linear programming.
Integer programming:
The variables in a program are restricted (in whole or in part) to integers and are called integer programs. When the variables in a linear model are restricted to integers, it is called integer linear programming. The currently popular methods for solving integer programming are often only applicable to integer linear programming. A class of mathematical programming that requires all or some of the variables in the solution of a problem to be integers. From the composition of constraints, it can be subdivided into linear, quadratic and nonlinear integer programming.
Dynamic programming:
Including knapsack problem, production and operation problem, fund management problem, resource allocation problem, shortest path problem and complex system reliability problem, etc.
Dynamic programming is mainly used to solve the optimization problem of the dynamic process divided into stages by time, but some static programming (such as linear programming and nonlinear programming) that have nothing to do with time can be regarded as a multi-stage decision-making process as long as the time factor is artificially introduced , can also be solved conveniently by dynamic programming method.
Multi-objective programming:
Multi-objective programming is a branch of mathematical programming. Study the optimization of more than one objective function over a given domain. Any multi-objective programming problem consists of two basic parts: