A comparison of gradient descent(green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information to take a more direct route. In calculus, Newton's method is an iterative method for finding the roots of a differentiable function f (i.e. solutions to the equation f(x)=0). In optimization, Newton's method is applied to the derivative f ′ of a twice-differentiable function f to find the roots of the derivative (solutions to f ′(x)=0), also known as the stationary points of f.
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% MATLAB code modified_newton.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% n_of_var -> number of design variables
% x = [-1.5 1.5] -> starting value of x
% epsilon1, epsilon2 -> constant used for terminating
% the algorithm
% delx -> required for gradient computation
% falpha_prev -> function valye at first / previous iteration
% deriv -> gradient vector
% sec_deriv -> hessian matrix
% search -> search direction (vector)
clear all
clc
n_of_var = 2;
x = [-3 2];
epsilon1 = 1e-7;
epsilon2 = 1e-7;
delx = 1e-3;
f_prev = func_multivar(x);
fprintf('Initial function value = %7.4f\n ',f_prev)
The Nelder–Mead method or downhill simplex method or amoeba method is a commonly applied numerical method used to find the minimum or maximum of an objective function in a multidimensional space. It is applied to nonlinear optimization problems for which derivatives may not be known. However, the Nelder–Mead technique is a heuristic search method that can converge to non-stationary points[1] on problems that can be solved by alternative methods.[2]The Nelder–Mead technique was proposed by John Nelder & Roger Mead (1965).[3]
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% MATLAB code neldermead.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% n_of_var -> number of design variables
% lb, ub -> lower/upper bound of variable
%(optinal for generating initial feasible points randomly)
% ybest -> best value of the objective function in the iteration
% ysecondbest -> second best value of the objective function
% yworst -> worst value of the objective function in the iteration
% xworst -> corresponding value of the variable for yworst
% xc -> centroid of the polygon
% fcentroid -> function value at xc
% deviation -> sum square deviation of function values from centroid
Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the positive of the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.
Gradient descent is also known as steepest descent, or the method of steepest descent. However, gradient descent should not be confused with the method of steepest descent for approximating integrals
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% MATLAB code steep_des.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% n_of_var -> number of design variables
% x = [-1.5 1.5] -> starting value of x
% epsilon1,epsilon2 ->constants used for terminating the algorithm
% delx -> required for gradient computation
% falpha_prev -> function valye at first / previous iteration
% deriv -> gradient vector
% search -> search direction (set to negative of gradient)
%
clear all
clc
n_of_var = 2;
x = [-3 2];
epsilon1 = 1e-6;
epsilon2 = 1e-6;
delx = 1e-3;
falpha_prev = func_multivar(x);
fprintf('Initial function value = %7.4f\n ',falpha_prev)
In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem (the duality principle). The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem.[1] However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.
In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem (the duality principle). The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem.[1] However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.
In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem (the duality principle). The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem.[1] However in general the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.[1][2][3][4][5] The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the twentieth century.[6]
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin.[7] Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial cones, and these become proper simplices with an additional constraint.[8][9][10][11] The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function.
The above function is known as 'cam' as described in L.C.W. Dixon and G.P. Szego (eds.), Towards Global Optimisation 2, North-Holland, Amsterdam, 1978.
Coding the Objective Function
We create a MATLAB file named simple_objective.m with the following code in it:
The Simulated Annealing solver assumes the objective function will take one input x where x has as many elements as the number of variables in the problem. The objective function computes the scalar value of the objective and returns it in its single return argument y.
Minimizing Using SIMULANNEALBND
To minimize our objective function using the SIMULANNEALBND function, we need to pass in a function handle to the objective function as well as specifying a start point as the second argument.
Optimization terminated: change in best function value less than options.TolFun.
x =
-0.0896 0.7130
fval =
-1.0316
exitFlag =
1
output =
iterations: 2948
funccount: 2971
message: 'Optimization terminated: change in best function value l...'
rngstate: [1x1 struct]
problemtype: 'unconstrained'
temperature: [2x1 double]
totaltime: 4.5000
The first two output arguments returned by SIMULANNEALBND are x, the best point found, and fval, the function value at the best point. A third output argument, exitFlag returns a flag corresponding to the reason SIMULANNEALBND stopped. SIMULANNEALBND can also return a fourth argument, output, which contains information about the performance of the solver.
Bound Constrained Minimization
SIMULANNEALBND can be used to solve problems with bound constraints. The lower and upper bounds are passed to the solver as vectors. For each dimension i, the solver ensures that lb(i) <= x(i) <= ub(i), where x is a point selected by the solver during simulation. We impose the bounds on our problem by specifying a range -64 <= x(i) <= 64 for x(i).
lb = [-64 -64];
ub = [64 64];
Now, we can rerun the solver with lower and upper bounds as input arguments.
fprintf('The number of iterations was : %d\n', output.iterations);
fprintf('The number of function evaluations was : %d\n', output.funccount);
fprintf('The best function value found was : %g\n', fval);
Optimization terminated: change in best function value less than options.TolFun.
The number of iterations was : 2428
The number of function evaluations was : 2447
The best function value found was : -1.03163
How Simulated Annealing Works
Simulated annealing mimics the annealing process to solve an optimization problem. It uses a temperature parameter that controls the search. The temperature parameter typically starts off high and is slowly "cooled" or lowered in every iteration. At each iteration a new point is generated and its distance from the current point is proportional to the temperature. If the new point has a better function value it replaces the current point and iteration counter is incremented. It is possible to accept and move forward with a worse point. The probability of doing so is directly dependent on the temperature. This unintuitive step sometime helps identify a new search region in hope of finding a better minimum.
An Objective Function with Additional Arguments
Sometimes we want our objective function to be parameterized by extra arguments that act as constants during the optimization. For example, in the previous objective function, say we want to replace the constants 4, 2.1, and 4 with parameters that we can change to create a family of objective functions. We can re-write the above function to take three additional parameters to give the new minimization problem.
min f(x) = (a - b*x1^2 + x1^4/3)*x1^2 + x1*x2 + (-c + c*x2^2)*x2^2;
x
a, b, and c are parameters to the objective function that act as constants during the optimization (they are not varied as part of the minimization). One can create a MATLAB file called parameterized_objective.m containing the following code.
function y = parameterized_objective(x,a,b,c)
y = (a - b*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + ...
(-c + c*x(2)^2)*x(2)^2;
Minimizing Using Additional Arguments
Again, we need to pass in a function handle to the objective function as well as a start point as the second argument.
SIMULANNEALBND will call our objective function with just one argument x, but our objective function has four arguments: x, a, b, c. We can use an anonymous function to capture the values of the additional arguments, the constants a, b, and c. We create a function handle 'ObjectiveFunction' to an anonymous function that takes one input x, but calls 'parameterized_objective' with x, a, b and c. The variables a, b, and c have values when the function handle 'ObjectiveFunction' is created, so these values are captured by the anonymous function.
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