Optimization Toolbox

## Key Features

Using a mixed-integer linear programming problem to determine the best way to supply sales locations from warehouses and factories.

## Defining and Solving Optimization Problems

Optimization techniques are used to find a set of design parameters or decisions that give the best possible result. An optimization problem is a model of a design or decision problem. You can model the design parameters or decisions as optimization variables. The other components of an optimization problem include objective functions and, if applicable, constraints. The objective function calculates the desired quantity to be minimized or maximized. Constraints limit the possible values for the optimization variables.

Least squares problems and nonlinear equation systems are optimization problems because they are solved by minimizing a sum of squares.

Transform a problem description into a mathematical program that can be solved using optimization, using a steam and electric power plant example.

### Problem-Based Optimization Modeling

Define your objective and constraints with the problem-based workflow using expressions built with the optimization variables. This approach can be used for linear and mixed-integer linear optimization problems.

Solve a linear program with Optimization Toolbox solvers and a problem-based approach, using a steam and electric power plant example.

### Solver-Based Optimization Modeling

Define your objective and constraints using the solver-based workflow. You can use functions or matrices with the optimization variables that your selected solver requires. This approach can be used with all the solvers in Optimization Toolbox™.

Solve a linear program using Optimization Toolbox solvers, using a steam and electric power plant example.

### Choosing a Solver

Optimization Toolbox contains different solvers for different types of objectives and constraints. The Optimization Decision Table helps you choose the best solver for your problem when using the solver-based approach. When using the problem-based approach, the appropriate solver is automatically selected for you.

Optimization Toolbox provides a broad range of functions so you can solve many types of optimization problems using a single function call. You can also use these functions to build a more complex optimization algorithm, such as a column generation algorithm, or in algorithms where optimization is one of the steps, such as motion planning for robots.

### Setting Options

Solver options enable you to tune or modify the optimization process and visualize solver progress.

Set options with optimoptions in Optimization Toolbox to tune solvers and monitor optimization progress.

### Supplying Derivatives

Optimization Toolbox solvers minimize nonlinear functions by estimating the partial derivatives of the objective function using finite differences. You can significantly reduce the overhead of the derivative estimation step by defining functions that calculate the values of the partial derivatives,

Calculating partial derivatives of an objective function can be tedious. You can use built-in functions for automatically calculating objective function partial derivatives by expressing the problem symbolically using Symbolic Math Toolbox™. You can then generate MATLAB® code that you can use with Optimization Toolbox solvers.

### Deployment

You can use Optimization Toolbox solvers with MATLAB Compiler™ to create decision support tools that can be shared with users who do not have MATLAB. These standalone applications can be deployed royalty-free to an unlimited number of end users. You can also integrate MATLAB optimization algorithms with other languages, such as C++, Java®, Python®, and .NET using MATLAB Compiler SDK™.

## Nonlinear Optimization

Optimization Toolbox provides widely used optimization algorithms for solving nonlinear programming problems in MATLAB. The toolbox includes solvers for unconstrained and constrained nonlinear optimization.

### Unconstrained Nonlinear Optimization

Optimization Toolbox uses three algorithms to solve unconstrained nonlinear minimization problems:

• The Quasi-Newton algorithm uses a mixed quadratic and cubic line search procedure and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula for updating the approximation of the Hessian matrix.
• The Nelder-Mead algorithm (or downhill simplex) is a direct-search algorithm that uses only function values (does not require derivatives) and handles nonsmooth objective functions. Global Optimization Toolbox provides additional derivative-free optimization algorithms for nonlinear optimization.
• The trust-region algorithm is used for unconstrained nonlinear problems and is especially useful for large-scale problems where sparsity or structure can be exploited.

### Constrained Nonlinear Optimization

Constrained nonlinear optimization problems are composed of linear or nonlinear objective functions and may be subject to linear and nonlinear constraints. Optimization Toolbox uses three algorithms to solve these problems:

• The interior-point algorithm is used for general nonlinear optimization. It is especially useful for large-scale problems that have sparsity or structure, and tolerates user-defined objective and constraint function evaluation failures. It is based on a barrier function, and optionally keeps all iterations strictly feasible with respect to bounds during the optimization run.
• The SQP algorithm is used for general nonlinear optimization. It honors bounds at all iterations and tolerates user-defined objective and constraint function evaluation failures.
• The trust-region reflective algorithm is used for bound constrained problems or linear equalities only. It is especially useful for large-scale problems.

The interior-point and trust-region reflective algorithms enable you to estimate Hessian matrices using different approaches.

For the interior-point algorithm, you can estimate Hessian matrices using:

• BFGS (dense)
• Limited memory BFGS (for large-scale problems)
• Hessian-multiply function
• Actual Hessian (sparse or dense)
• Finite difference of gradients, without requiring knowledge of sparsity structure

For the trust-region reflective algorithm, you can use:

• Finite difference of gradients, sparsity structure of the Hessian
• Actual Hessian (sparse or dense)
• Hessian-multiply function

To lower memory usage, the interior-point and trust-region reflective algorithms enable you to calculate Hessian-times-vector products in a function without having to form the Hessian matrix explicitly.

Optimization Toolbox can solve large-scale linear and quadratic programming problems.

### Linear Programming

Linear programming problems involve minimizing or maximizing a linear objective function subject to bounds, linear equality, and inequality constraints. Linear programming is used in finance, energy, operations research, and other applications where relationships between variables can be expressed linearly.

Optimization Toolbox includes two algorithms used to solve linear programming problems:

• The dual-simplex algorithm is a type of simplex algorithm. Simplex algorithms are systematic procedures for generating and testing candidate vertex solutions to a linear program and are the most widely used algorithms for linear programming.
• The interior-point algorithm is based on a primal-dual predictor-corrector algorithm used for solving linear programming problems. Interior-point is especially useful for large-scale problems that have structure or can be defined using sparse matrices.

Quadratic programming problems involve minimizing a multivariate quadratic function subject to bounds, linear equality, and inequality constraints. Quadratic programming is used for portfolio optimization in finance, power generation optimization for electrical utilities, design optimization in engineering, and other applications.

Optimization Toolbox includes two algorithms for solving quadratic programs:

• The interior-point-convex algorithm solves convex problems with any combination of constraints.
• The trust-region-reflective algorithm solves bound constrained problems or linear equality constrained problems.
In this webinar, you will learn how MATLAB can be used to solve optimization problems using an example quadratic optimization problem and the symbolic math tools in MATLAB.

Both the interior-point-convex and trust-region-reflective algorithms are large scale, meaning they can handle large, sparse problems. Furthermore, the interior-point-convex algorithm has optimized internal linear algebra routines and a presolve module that can improve speed, numerical stability, and the detection of infeasibility.

## Mixed-Integer Linear Programming

Mixed-integer linear programming expands the linear programming problem with the additional constraint that some or all of the variables in the optimal solution must be integers.

For some optimization problems, the variables should not take on fractional values. For instance, if a variable represents the number of stock shares to purchase, it should take on only integer values. Similarly, if a variable represents the on/off state of a generator, it should take on only binary values (0 or 1). The mixed-integer linear programming problem allows this behavior to be modeled by adding the constraint that these variables should take on only integers, or whole numbers, in the optimal solution.

Through a steel blending example, you will learn how to solve a mixed-integer linear program using Optimization Toolbox solvers and a problem-based approach.

Optimization Toolbox solves mixed-integer linear programming problems using an algorithm that:

• Performs integer programming preprocessing to tighten the feasible region
• Applies cutting planes to tighten the feasible region
• Uses heuristics to search for integer feasible solutions
• Verifies that no better feasible solution is possible with a branch and bound algorithm that solves a series of linear programming relaxation problems

## Multiobjective Optimization

Multiobjective optimization is concerned with the minimization of multiple objective functions that are subject to a set of constraints. Optimization Toolbox provides functions for solving two formulations of multiobjective optimization problems:

• The goal attainment problem involves reducing the value of a linear or nonlinear vector function to attain the goal values given in a goal vector. The relative importance of the goals is indicated using a weight vector. The goal attainment problem may also be subject to linear and nonlinear constraints.
• The minimax problem involves minimizing the worst-case value of a set of multivariate functions, possibly subject to linear and nonlinear constraints.

Optimization Toolbox transforms both types of multiobjective problems into standard constrained optimization problems and then solves them using an active-set approach.

Global Optimization Toolbox provides an additional multiobjective solver for nonsmooth problems.

## Nonlinear Least Squares, Data Fitting, and Nonlinear Equations

Optimization Toolbox can solve linear and nonlinear least-squares problems, data fitting problems, and nonlinear equations.

### Linear and Nonlinear Least-Squares Optimization

The toolbox includes two algorithms for solving constrained linear least-squares problems:

• The interior-point algorithm solves problems with bounds and linear inequalities or equalities.
• The trust-region-reflective algorithm solves large-scale problems that have only bound constraints.

The toolbox includes two algorithms for solving nonlinear least-squares problems:

• The trust-region-reflective algorithm implements a trust-region approach specialized to nonlinear least squares problems. It is used for unconstrained and bound-constrained problems.
• The Levenberg-Marquardt algorithm is used for unconstrained problems.

### Data Fitting

The toolbox provides a specialized interface for data fitting problems in which you want to find the member of a family of nonlinear functions that best fits a set of data points. The toolbox uses the same algorithms for data fitting problems that it uses for nonlinear least-squares problems.

### Nonlinear Equation Solving

Optimization Toolbox implements a dogleg trust-region algorithm for solving a system of nonlinear equations where there are as many equations as unknowns. The toolbox can also solve this problem using trust-region and Levenberg-Marquardt algorithms.

## Parallel Computing

Optimization Toolbox can be used with Parallel Computing Toolbox to solve problems that benefit from parallel computation. You can decrease time to solution by enabling built-in parallel computing support or by defining a custom parallel computing implementation of an optimization problem.

Built-in support for parallel computing in Optimization Toolbox enables you to accelerate the gradient estimation step in solvers for general nonlinear optimization problems, nonlinear least squares, nonlinear systems of equations, and for multiobjective problems.

You can customize a parallel computing implementation by explicitly defining the optimization problem to use parallel computing functionality. You can define either an objective or a constraint function to use parallel computing, enabling you to decrease the time required to evaluate the objective or constraint.

In this webinar, we will use two case studies to demonstrate how you can use parallel computing to speed up single-level and multilevel optimization problems in MATLAB.