【Advanced Chapter】MATLAB Mathematical Optimization Toolbox: Optimization Toolbox User Guide

# 2.1 Linear Programming (LP)

### 2.1.1 LP Models and Solution Methods

Linear Programming (LP) is a mathematical optimization technique used to solve optimization problems with linear objective functions and linear constraints. The LP model can be represented as:

max/min f(x) = c^T x
subject to:
Ax ≤ b
x ≥ 0

where:

- `f(x)`: The objective function
- `x`: Decision variables
- `c`: Objective function coefficient vector
- `A`: Constraint matrix
- `b`: Constraint value vector

LP problems can be solved using algorithms such as the simplex method or interior-point method. The simplex method is an iterative algorithm that starts from a feasible solution and gradually approaches the optimal solution through a series of steps. The interior-point method is a direct method that iteratively searches for the optimal solution within the feasible domain.

### 2.1.2 Applications of LP in Real-World Problems

LP has a wide range of applications in real-world problems, including:

- Resource allocation: Allocate limited resources to maximize or minimize the objective function (e.g., profit, cost).
- Production planning: Determine production plans to meet demand and optimize costs.
- Portfolio optimization: Allocate investments to maximize returns and minimize risks.
- Transportation problems: Optimize the transportation routes of goods to minimize costs or time.

# 2. Optimization Theory and Algorithms

### 2.1 Linear Programming (LP)

#### 2.1.1 LP Models and Solution Methods

**LP Model**

Linear Programming (LP) is an optimization problem where the objective function and constraints are linear. The general form of an LP model is as follows:

max/min f(x) = c^T x
subject to:
Ax ≤ b
x ≥ 0

where:

* f(x) is the objective function to be maximized or minimized
* x is the vector of decision variables
* c is the vector of objective function coefficients
* A is the constraint matrix
* b is the vector of constraint values

**Solution Methods**

LP problems can be solved using a variety of algorithms, including:

***Simplex Method:** An iterative algorithm that starts from a feasible solution and gradually moves to better solutions until the optimal solution is reached.
***Interior-Point Method:** A non-iterative algorithm that starts from within the feasible domain and moves directly towards the optimal solution.

#### 2.1.2 Applications of LP in Real-World Problems

LP is widely applied in real-world problems, including:

***Resource Allocation:** Allocate limited resources to maximize or minimize the objective function (e.g., profit, cost).
***Production Planning:** Determine production plans to maximize output or minimize costs.
***Transportation Problems:** Optimize the transportation of goods from multiple sources to multiple destinations.
***Portfolio Optimization:** Allocate investments to maximize returns or minimize risks.

### 2.2 Nonlinear Programming (NLP)

#### 2.2.1 NLP Models and Solution Methods

**NLP Model**

Nonlinear Programming (NLP) is an optimization problem where the objective function or constraints are nonlinear. The general form of an NLP model is as follows:

max/min f(x)
subject to:
h(x) ≤ 0
g(x) = 0

where:

* f(x) is the objective function
* x is the vector of decision variables
* h(x) is the vector of inequality constraints
* g(x) is the vector of equality constraints

**Solution Methods**

NLP problems can be solved using a variety of algorithms, including:

***Gradient Descent Method:** An iterative algorithm that moves in the direction of the negative gradient until a local optimum is reached.
***Newton's Method:** An iterative algorithm that uses gradient and Hessian matrix information to approach the optimal solution at a quadratic convergence rate.
***Interior-Point Method:** A non-iterative algorithm that starts from within the feasible domain and moves directly towards the optimal solution.

#### 2.2.2 Applications of NLP in Real-World Problems

NLP is also widely applied in real-world problems, including:

***Engineering Design:** Optimize the design of structures, fluid dynamics, and thermodynamic systems.
***Financial Modeling:** Optimize portfolios, risk management, and pricing models.
***Data Analysis:** Optimize machine learning models, data mining, and predictive algorithms.

### 2.3 Integer Programming (IP)

#### 2.3.1 IP Models and Solution Methods

**IP Model**

Integer Programming (IP) is an optimization problem where the decision variables must take integer values. The general form of an IP model is as follows:

max/min f(x)
subject to:
Ax ≤ b
x ≥ 0
x ∈ Z^n

where:

* f(x) is the objective function
* x is the vector of decision variables
* A is the constraint matrix
* b is the constraint value vector
* Z^n is the set of integer values for decision variables

**Solution Methods**

IP problems can be solved using a variety of algorithms, including:

***Branch and Bound Method:** A recursive algorithm that decomposes the problem into subproblems and uses branching and bounding techniques to find the optimal solution.
***Cutting Plane Method:** An iterative algorithm that tightens the feasible region by adding constraints to approach the optimal solution.

#### 2.3.2 Applications of IP in Real-World Problems

IP has a wide range of applications in real-world problems, including:

***Scheduling Problems:** Optimize the scheduling of personnel, equipment, and resources to maximize efficiency or minimize costs.
***Facility Location:** Determine the location of facilities to minimize transportation costs or maximize customer coverage.
***Network Optimization:** Optimize the flow in a network to maximize throughput or minimize delay.

# 3.1 Basic Functions and Syntax

#### 3.1.1 Definition and Solution of Optimization Problems

MATLAB Optimization Toolbox provides a set of functions to define and solve optimization problems. The core function is `fminunc`