# Optimization in MATLAB: Video Tutorial

URL: https://yarpiz.com/optimization-video-tutorial-yptnm190520s20
Author: Mostapha Kalami Heris
Published: 2019-07-03
Updated: 2026-10-30
Topics: Courses

> Model linear, integer, quadratic and nonlinear optimization problems in MATLAB. Follow worked examples and check feasibility, solver termination and local solutions.

This video tutorial reviews optimization and shows how to implement it in MATLAB. It is part of the [Numerical Computations course](https://yarpiz.com/numerical-computations-in-matlab-video-tutorial-572).

## Optimization problems in MATLAB

An optimization problem asks for values of the decision variables $x$ that improve an objective function $f(x)$, subject to constraints. The solvers used in this tutorial minimize. To maximize a function with these solvers, minimize its negative and change the sign of the returned objective value.

The three videos follow the type of the objective and the constraints:

- **Linear programming (LP):** a linear objective with linear constraints, solved with `linprog`. If some variables must be integers, the problem is a mixed integer linear program (MILP), solved with `intlinprog`.
- **Quadratic programming (QP):** a quadratic objective with linear constraints, solved with `quadprog`.
- **Nonlinear optimization:** a general objective, with or without constraints, solved with `fminbnd`, `fminsearch`, `fminunc` or `fmincon`.

`linprog`, `intlinprog`, `quadprog`, `fminunc` and `fmincon` are part of Optimization Toolbox. `fminbnd` and `fminsearch` are in core MATLAB.

## Watch online

The three sections of this tutorial are on YouTube and are embedded here as a playlist.

::: video {playlist="PLgH3sgdvgO4SDFQNSoM-YlYupGFlXuWOE" youtube="kRv7QMUH7gA" count="3" title="Optimization in MATLAB: Video Tutorial"}
:::

## What each video covers

### Section 1: Linear programming and mixed integer LP

Linear Programming and Mixed-Integer LP ([YouTube](https://youtu.be/kRv7QMUH7gA)). MATLAB writes a linear program in this standard form:

$$
\min_{x}\; f^{T}x \quad \text{subject to} \quad A x \le b, \quad A_{eq}\, x = b_{eq}, \quad lb \le x \le ub
$$

So a model is defined by seven matrices and vectors: `f`, `A`, `b`, `Aeq`, `beq`, `lb` and `ub`. Missing parts are passed as empty matrices `[]`.

The video solves three examples:

1. Minimize $2x_1 - 5x_2$ with $100 \le x_1 \le 200$, $50 \le x_2 \le 150$ and $x_1 + x_2 \le 210$. The bounds go into `lb` and `ub`, and `linprog` returns an objective value of $-350$.
2. Maximize $x_1 + x_2/3$ subject to six linear inequalities. The objective is negated, and each "greater than or equal" constraint is multiplied by $-1$ to fit the form $Ax \le b$. The solution is $x = (2/3, 4/3)$ with a maximum of about $1.111$. The video then adds integer constraints with `intlinprog`, whose second input `intcon` lists the integer variables. With $x_1$ an integer the solution is $(1, 0)$, and with $x_2$ an integer it is $(0.75, 1)$.
3. Minimize $5x_1 - 2x_2$ subject to $x_1 - 3x_2 \ge -14$, $4x_1 - x_2 \ge 33$, $x_1 - 2x_2 \le 25$ and $x \ge 0$. The LP solution is about $(10.27, 8.09)$. With $x_2$ an integer, `intlinprog` returns $(10.25, 8)$.

```matlab {title="Solve a mixed-integer linear program"}
f = [5; -2];
A = [-1 3; -4 1; 1 -2];
b = [14; -33; 25];
lb = [0; 0];
x = intlinprog(f, 2, A, b, [], [], lb)
```

For a longer treatment of `linprog`, with the transportation problem as the example, see the [Linear Programming in MATLAB video tutorial](https://yarpiz.com/linear-programmin-in-matlab-linprog-yptlp190516).

### Section 2: Quadratic programming

Quadratic Programming ([YouTube](https://youtu.be/XBA_VemmaxY)). A quadratic program adds a quadratic term to the LP form:

$$
\min_{x}\; \tfrac{1}{2}\, x^{T} H x + f^{T}x \quad \text{subject to} \quad A x \le b, \quad A_{eq}\, x = b_{eq}, \quad lb \le x \le ub
$$

The video notes that training a support vector machine leads to a QP. Its example is

$$
\min\; \tfrac{1}{2}x_1^2 + x_2^2 - x_1 x_2 - 2x_1 - 6x_2 \quad \text{subject to} \quad x_1 + x_2 \le 2, \quad 2x_1 + x_2 \le 5
$$

The main step is to find $H$. The objective is written as $\tfrac{1}{2}(x_1^2 + 2x_2^2 - 2x_1x_2)$, and the cross term is split equally between the two off-diagonal elements, because a symmetric $H$ is preferred:

```matlab {title="Solve a constrained quadratic program"}
H = [1 -1; -1 2];
f = [-2; -6];
A = [1 1; 2 1];
b = [2; 5];
[x, fval] = quadprog(H, f, A, b)
```

The solution is $x = (0.4, 1.6)$ with an objective value of $-8.4$.

### Section 3: Constrained and unconstrained nonlinear optimization

Constrained and Unconstrained Nonlinear Optimization ([YouTube](https://youtu.be/7nCgLN00yqk)). The video compares four functions:

- `fminbnd`: one variable, searched within an interval.
- `fminsearch`: several variables, no constraints. It searches without requiring derivatives, and the video uses it for a small problem. This does not guarantee reliable behavior for discontinuous functions or large problems.
- `fminunc`: several variables, no constraints. The video uses it for smooth objectives, for which derivative information is useful.
- `fmincon`: several variables with constraints.

The first example is $f(x) = x^3 - 5x + 1$. `fminbnd` on $[-1, 5]$ finds the local minimum at about $1.29$, and `fminsearch` and `fminunc` started from 2 find the same point. The video also turns root finding into optimization: minimizing $|f(x)|$ with `fminbnd` on different intervals finds each of the three roots, which `roots([1 0 -5 1])` confirms.

The second example is the Rosenbrock function

$$
f(x) = 100\,(x_2 - x_1^2)^2 + (1 - x_1)^2
$$

whose minimum is at $(1, 1)$. Both `fminsearch` and `fminunc`, started from $(0, 0)$, find it.

The last example is a constrained design problem: a box with sides $x_1$, $x_2$ and $x_3$ should have the largest volume, while its surface area is at most 150:

$$
\max\; x_1 x_2 x_3 \quad \text{subject to} \quad 2\,(x_1 x_2 + x_1 x_3 + x_2 x_3) \le 150, \quad x \ge 0
$$

The nonlinear constraint goes into a function that returns `c` (inequalities, $c(x) \le 0$) and `ceq` (equalities, empty here). Since `fmincon` minimizes, the objective is the negative volume. Started from $(1, 1, 1)$, the solution is a cube with all sides equal. Adding the constraint $2x_2 \le x_3$ gives a box that is no longer a cube, and the volume drops to about 117. The video adds this constraint to `c`, and notes that a linear constraint can also go into `A` and `b`.

## Applications

- Planning and allocation problems with linear models, such as production planning and transportation.
- Problems with yes or no decisions or whole units, modeled with integer variables.
- Machine learning models such as support vector machines (quadratic programming).
- Engineering design, such as the box example, where the objective and constraints are nonlinear.

## Check feasibility and optimality

Substitute the result into every constraint and evaluate the original objective, especially after negating a maximization problem. Inspect the solver's termination information before accepting a result. A local solution of a nonlinear problem need not be the global best, so compare starting points when the model has several basins. Keep integer decisions in an integer model rather than rounding a continuous solution, which can break the constraints.

## How to cite

Mostapha Kalami Heris, Optimization in MATLAB: Video Tutorial (YPTNM190520S20), Yarpiz, 2019. https://yarpiz.com/optimization-video-tutorial-yptnm190520s20