O método Quase-Newton é baseado em regras recursivas para estimar a matriz Hessiana inversa. Dois métodos eficientes são o DFP e o BFGS, que fazem parte da Família de Broyden, a qual utiliza uma combinação linear dos métodos DFP e BFGS para a estimativa. O código fornece a implementação do método Quase-Newton em MATLAB.
6. Método SIMPLEX:
1) Introduz o método simplex para resolver problemas de programação linear, incluindo colocar o problema na forma padrão e implementar o algoritmo simplex.
2) Discutem os tipos de soluções que podem surgir ao resolver um problema de programação linear usando o método simplex, como solução única ótima, múltiplas soluções ótimas e solução ilimitada.
3) Apresentam o método das duas fases para lidar com problemas que não tenham solução ou tenham solução ilimitada.
O documento descreve o uso dos métodos de Newton e da Bissecção para encontrar as raízes de um polinômio de Bessel. Códigos em MATLAB são fornecidos para implementar os métodos e gerar gráficos das soluções. O método da Bissecção requer mais iterações mas é mais eficaz para encontrar a raiz, embora o erro seja maior do que para o método de Newton. O método da Bissecção é mais indicado para este polinômio.
Plano de aula po1 capitulo 6 método simplex 2015 vrs 0000 - fazer.ppt [modo...Luis Duncan
O documento descreve o método Simplex para resolver problemas de programação linear (PPL), incluindo como colocar um PPL na forma padrão e implementar o algoritmo Simplex. O método Simplex envolve colocar o PPL em uma forma matricial adequada e, em seguida, iterar por meio de quadros para otimizar a função objetivo, resultando em uma solução ótima ou não para o PPL.
Plano de aula po1 capitulo 4 método gráfico 2015 vrs 0000.ppt [modo de comp...Luis Duncan
Este documento discute o método gráfico para resolver problemas de programação linear (PPL) com duas variáveis de decisão. Apresenta as características e limitações do método gráfico, como a interpretação no plano cartesiano R2 e tipos de soluções possíveis de um PPL, como solução única, múltiplas soluções, solução ilimitada, sem solução e degenerada. Exemplos ilustram cada tipo de solução e exercícios são fornecidos para aplicar o método gráfico.
This document describes an in-class activity where students work in teams to optimize different bivariate functions using the optim() function in R. It provides instructions for students to claim a test function, plot it, and use the Nelder-Mead and simulated annealing methods to optimize the function. Several student examples are given, including optimizing the sphere, Rosenbrock, Beale's and Goldstein-Price functions. Errors are reported for some optimizations.
This document provides an overview of optimization techniques. It defines optimization as identifying variable values that minimize or maximize an objective function subject to constraints. It then discusses various applications of optimization in finance, engineering, and data modeling. The document outlines different types of optimization problems and algorithms. It provides examples of unconstrained optimization algorithms like gradient descent, conjugate gradient, Newton's method, and BFGS. It also discusses the Nelder-Mead simplex algorithm for constrained optimization and compares the performance of these algorithms on sample problems.
El documento resume las últimas novedades en educación y formación profesional en Europa, incluyendo el aprendizaje permanente y las sociedades en red. También describe las visitas de la asociación HETEL a centros de formación profesional en Suecia, y la participación de HETEL en un seminario europeo sobre tutorización en línea y educación a distancia. Además, presenta novedades sobre varios centros de formación en el País Vasco.
Quasi-Newton methods approximate the inverse Hessian matrix to help solve unconstrained optimization problems more efficiently. The update matrix is approximated and updated at each iteration based on the gradient vectors. The most widely used update formulas are the Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno formulas, which use symmetric rank-two updates to iteratively improve the inverse Hessian approximation. While quasi-Newton methods can converge faster than gradient descent, they may fail to find the optimal solution for non-quadratic problems.
6. Método SIMPLEX:
1) Introduz o método simplex para resolver problemas de programação linear, incluindo colocar o problema na forma padrão e implementar o algoritmo simplex.
2) Discutem os tipos de soluções que podem surgir ao resolver um problema de programação linear usando o método simplex, como solução única ótima, múltiplas soluções ótimas e solução ilimitada.
3) Apresentam o método das duas fases para lidar com problemas que não tenham solução ou tenham solução ilimitada.
O documento descreve o uso dos métodos de Newton e da Bissecção para encontrar as raízes de um polinômio de Bessel. Códigos em MATLAB são fornecidos para implementar os métodos e gerar gráficos das soluções. O método da Bissecção requer mais iterações mas é mais eficaz para encontrar a raiz, embora o erro seja maior do que para o método de Newton. O método da Bissecção é mais indicado para este polinômio.
Plano de aula po1 capitulo 6 método simplex 2015 vrs 0000 - fazer.ppt [modo...Luis Duncan
O documento descreve o método Simplex para resolver problemas de programação linear (PPL), incluindo como colocar um PPL na forma padrão e implementar o algoritmo Simplex. O método Simplex envolve colocar o PPL em uma forma matricial adequada e, em seguida, iterar por meio de quadros para otimizar a função objetivo, resultando em uma solução ótima ou não para o PPL.
Plano de aula po1 capitulo 4 método gráfico 2015 vrs 0000.ppt [modo de comp...Luis Duncan
Este documento discute o método gráfico para resolver problemas de programação linear (PPL) com duas variáveis de decisão. Apresenta as características e limitações do método gráfico, como a interpretação no plano cartesiano R2 e tipos de soluções possíveis de um PPL, como solução única, múltiplas soluções, solução ilimitada, sem solução e degenerada. Exemplos ilustram cada tipo de solução e exercícios são fornecidos para aplicar o método gráfico.
This document describes an in-class activity where students work in teams to optimize different bivariate functions using the optim() function in R. It provides instructions for students to claim a test function, plot it, and use the Nelder-Mead and simulated annealing methods to optimize the function. Several student examples are given, including optimizing the sphere, Rosenbrock, Beale's and Goldstein-Price functions. Errors are reported for some optimizations.
This document provides an overview of optimization techniques. It defines optimization as identifying variable values that minimize or maximize an objective function subject to constraints. It then discusses various applications of optimization in finance, engineering, and data modeling. The document outlines different types of optimization problems and algorithms. It provides examples of unconstrained optimization algorithms like gradient descent, conjugate gradient, Newton's method, and BFGS. It also discusses the Nelder-Mead simplex algorithm for constrained optimization and compares the performance of these algorithms on sample problems.
El documento resume las últimas novedades en educación y formación profesional en Europa, incluyendo el aprendizaje permanente y las sociedades en red. También describe las visitas de la asociación HETEL a centros de formación profesional en Suecia, y la participación de HETEL en un seminario europeo sobre tutorización en línea y educación a distancia. Además, presenta novedades sobre varios centros de formación en el País Vasco.
Quasi-Newton methods approximate the inverse Hessian matrix to help solve unconstrained optimization problems more efficiently. The update matrix is approximated and updated at each iteration based on the gradient vectors. The most widely used update formulas are the Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno formulas, which use symmetric rank-two updates to iteratively improve the inverse Hessian approximation. While quasi-Newton methods can converge faster than gradient descent, they may fail to find the optimal solution for non-quadratic problems.
A simplex nelder mead genetic algorithm for minimizing molecular potential en...Aboul Ella Hassanien
The document describes a new genetic Nelder-Mead algorithm (GNMA) for minimizing molecular potential energy functions. GNMA uses a population partitioning approach and applies genetic operations to submatrices before using Nelder-Mead optimization on elite solutions. It was tested on a simplified molecular potential energy function and compared to 9 other algorithms, showing promising performance with less computational expense than other methods. The authors conclude GNMA is an efficient approach for molecular structure optimization problems.
This document discusses optimization techniques in R. It provides an overview of commonly used optimization functions like optim and optimize in R. It also discusses different optimization algorithms like conjugate gradient (CG), simulated annealing (SANN), Broyden–Fletcher–Goldfarb–Shanno (BFGS) and Nelder-Mead methods that are used in optim. The document provides examples of parameter estimation using optim and compares the results with linear regression. It also discusses applying optimization for problems like maximum likelihood estimation.
The document presents a comparative study of algorithms for nonlinear optimization. It evaluates several nonlinear optimization algorithms (conjugate gradient methods, quasi-Newton methods, Powell's method) on standard test functions (Booth, Himmelblau, Beale) from different initial points and step sizes. For the Booth function, all algorithms converge in 2 steps. For Himmelblau, Polak-Ribiere and BFGS methods perform best. For Beale, conjugate gradient methods converge for some initial points but not all, while Powell's method converges from more points.
The Nelder-Mead search algorithm is an optimization technique used to find the minimum or maximum of an objective function by iteratively modifying a set of parameter values. It begins with an initial set of points forming a simplex in parameter space and evaluates the objective function at each point, replacing the highest point with a new point to form a new simplex. This process repeats until the optimal value is found or a stopping criteria is met. The algorithm is simple to implement and can find local optima for problems with multiple parameters, but depends on the initial starting point and may get stuck in local optima rather than finding the global solution.
This curriculum vitae outlines the career and accomplishments of Tarun K. Dhar, a scientist and head of the Drug Development, Diagnostics and Biotechnology Division at the Indian Institute of Chemical Biology in Kolkata, India. It details his education, awards, professional appointments rising to the position of Scientist G, research interests in developing new immunochemical methods and immunoassays, notable research contributions including developing new signal amplification technologies for ELISA, publications, patents, and professional affiliations.
Generalized linear models (GLMs) are a well-known statistical method for fitting linear models to different types of prediction problems. GLMs allow modeling of non-normal distributions and non-identity link functions. Regularization techniques like lasso, ridge, and elastic net are commonly used with GLMs to prevent overfitting. H2O's GLM implementation handles large datasets with many predictors efficiently using distributed and parallel processing. It supports automatic handling of categorical variables, multiple solvers, and regularization including elastic net. GLMs scaled to "big data" problems can be fit in seconds on H2O using all CPU cores.
This document discusses various concepts related to simulated annealing including the acceptance function, initial temperature, equilibrium state, cooling schedule, stopping condition, and handling constraints. It describes how the acceptance of non-improving moves is based on temperature and change in objective function. It also provides examples of different cooling schedules and discusses how to determine equilibrium state and stopping criteria. The document concludes with applying simulated annealing to solve the knapsack problem.
Using Gradient Descent for Optimization and LearningDr. Volkan OBAN
This document discusses optimization techniques for gradient descent, including the basics of gradient descent, Newton's method, and quasi-Newton methods. It covers limitations of gradient descent and Newton's method, and approximations like Gauss-Newton, Levenberg-Marquardt, BFGS, and L-BFGS. It also discusses stochastic optimization techniques for handling large datasets with minibatch or online updates rather than full batch updates.
Simulated Annealing - A Optimisation TechniqueAUSTIN MOSES
Simulated annealing is a global optimization technique inspired by the physical process of annealing in solids. It can find the global minimum of a cost function by slowly cooling the system. At each temperature, the algorithm accepts random moves to neighboring solutions with a probability based on the change in cost and current temperature. This allows occasionally moving to higher-cost solutions and avoids getting stuck in local minima. While slower than local search methods, simulated annealing is more likely to find the global optimum solution over multiple iterations as the temperature decreases.
SA is a global optimization technique.
It distinguishes between different local optima.
It is a memory less algorithm & the algorithm does not use any information gathered during the search.
SA is motivated by an analogy to annealing in solids.
& it is an iterative improvement algorithm.
Simulated annealing -a informative approachRanak Ghosh
Simulated annealing is an optimization algorithm inspired by the annealing process in metals. It can be used to find approximate global minima in large search spaces. The algorithm works as follows: initially random solutions are accepted probabilistically, with a higher probability of accepting worse solutions at higher "temperatures"; the temperature is slowly decreased, making the probability of accepting worse solutions decrease as well, until eventually only improvements are accepted. This process allows the algorithm to avoid getting stuck in local minima. Simulated annealing has been successfully applied to problems like circuit placement, scheduling, and the traveling salesperson problem. It provides good quality solutions efficiently compared to other optimization techniques.
A simplex nelder mead genetic algorithm for minimizing molecular potential en...Aboul Ella Hassanien
The document describes a new genetic Nelder-Mead algorithm (GNMA) for minimizing molecular potential energy functions. GNMA uses a population partitioning approach and applies genetic operations to submatrices before using Nelder-Mead optimization on elite solutions. It was tested on a simplified molecular potential energy function and compared to 9 other algorithms, showing promising performance with less computational expense than other methods. The authors conclude GNMA is an efficient approach for molecular structure optimization problems.
This document discusses optimization techniques in R. It provides an overview of commonly used optimization functions like optim and optimize in R. It also discusses different optimization algorithms like conjugate gradient (CG), simulated annealing (SANN), Broyden–Fletcher–Goldfarb–Shanno (BFGS) and Nelder-Mead methods that are used in optim. The document provides examples of parameter estimation using optim and compares the results with linear regression. It also discusses applying optimization for problems like maximum likelihood estimation.
The document presents a comparative study of algorithms for nonlinear optimization. It evaluates several nonlinear optimization algorithms (conjugate gradient methods, quasi-Newton methods, Powell's method) on standard test functions (Booth, Himmelblau, Beale) from different initial points and step sizes. For the Booth function, all algorithms converge in 2 steps. For Himmelblau, Polak-Ribiere and BFGS methods perform best. For Beale, conjugate gradient methods converge for some initial points but not all, while Powell's method converges from more points.
The Nelder-Mead search algorithm is an optimization technique used to find the minimum or maximum of an objective function by iteratively modifying a set of parameter values. It begins with an initial set of points forming a simplex in parameter space and evaluates the objective function at each point, replacing the highest point with a new point to form a new simplex. This process repeats until the optimal value is found or a stopping criteria is met. The algorithm is simple to implement and can find local optima for problems with multiple parameters, but depends on the initial starting point and may get stuck in local optima rather than finding the global solution.
This curriculum vitae outlines the career and accomplishments of Tarun K. Dhar, a scientist and head of the Drug Development, Diagnostics and Biotechnology Division at the Indian Institute of Chemical Biology in Kolkata, India. It details his education, awards, professional appointments rising to the position of Scientist G, research interests in developing new immunochemical methods and immunoassays, notable research contributions including developing new signal amplification technologies for ELISA, publications, patents, and professional affiliations.
Generalized linear models (GLMs) are a well-known statistical method for fitting linear models to different types of prediction problems. GLMs allow modeling of non-normal distributions and non-identity link functions. Regularization techniques like lasso, ridge, and elastic net are commonly used with GLMs to prevent overfitting. H2O's GLM implementation handles large datasets with many predictors efficiently using distributed and parallel processing. It supports automatic handling of categorical variables, multiple solvers, and regularization including elastic net. GLMs scaled to "big data" problems can be fit in seconds on H2O using all CPU cores.
This document discusses various concepts related to simulated annealing including the acceptance function, initial temperature, equilibrium state, cooling schedule, stopping condition, and handling constraints. It describes how the acceptance of non-improving moves is based on temperature and change in objective function. It also provides examples of different cooling schedules and discusses how to determine equilibrium state and stopping criteria. The document concludes with applying simulated annealing to solve the knapsack problem.
Using Gradient Descent for Optimization and LearningDr. Volkan OBAN
This document discusses optimization techniques for gradient descent, including the basics of gradient descent, Newton's method, and quasi-Newton methods. It covers limitations of gradient descent and Newton's method, and approximations like Gauss-Newton, Levenberg-Marquardt, BFGS, and L-BFGS. It also discusses stochastic optimization techniques for handling large datasets with minibatch or online updates rather than full batch updates.
Simulated Annealing - A Optimisation TechniqueAUSTIN MOSES
Simulated annealing is a global optimization technique inspired by the physical process of annealing in solids. It can find the global minimum of a cost function by slowly cooling the system. At each temperature, the algorithm accepts random moves to neighboring solutions with a probability based on the change in cost and current temperature. This allows occasionally moving to higher-cost solutions and avoids getting stuck in local minima. While slower than local search methods, simulated annealing is more likely to find the global optimum solution over multiple iterations as the temperature decreases.
SA is a global optimization technique.
It distinguishes between different local optima.
It is a memory less algorithm & the algorithm does not use any information gathered during the search.
SA is motivated by an analogy to annealing in solids.
& it is an iterative improvement algorithm.
Simulated annealing -a informative approachRanak Ghosh
Simulated annealing is an optimization algorithm inspired by the annealing process in metals. It can be used to find approximate global minima in large search spaces. The algorithm works as follows: initially random solutions are accepted probabilistically, with a higher probability of accepting worse solutions at higher "temperatures"; the temperature is slowly decreased, making the probability of accepting worse solutions decrease as well, until eventually only improvements are accepted. This process allows the algorithm to avoid getting stuck in local minima. Simulated annealing has been successfully applied to problems like circuit placement, scheduling, and the traveling salesperson problem. It provides good quality solutions efficiently compared to other optimization techniques.
1. Método Quase-Newton
O método Quase-Newton é baseado em regras recursivas para formação de
uma matriz Hessiana (Hk) que é correspondente a uma estimativa de sua inversa.
‘Deve-se ter em mente que é obrigatório a matriz Hk seja sempre positiva.
Com as características citadas anteriormente foram definidos dois métodos
eficientes: DFP (Davidon-Fletcher-Powell) e o método BFGS (Broyden-Fletcher-
Goldfarb-Shanno) em homenagem aos seus formuladores. Existe, no entanto, uma
conexão entre esses dois métodos e criou-se então a Família de Broyden que contém
os dois métodos agrupados.
Método DFP
Observa-se o método DFP a seguir:
𝐶 𝑘
𝐷𝐹𝑃
= (𝑣 𝑘 𝑣 𝑘′/ 𝑣′ 𝑘 𝑟𝑘) – (𝐻 𝑘 𝑟𝑘 𝑟′ 𝑘 𝐻𝑘 /𝑟′ 𝑘 𝐻 𝑘 𝑟𝑘)
Método BFGS
O método BFGS é mostrado abaixo:
𝐶 𝑘
𝐵𝐹𝐺𝑆
= (1 + (𝑟′ 𝑘 𝐻 𝑘 𝑟𝑘/𝑟′ 𝑘 𝑣 𝑘) *
(𝑣 𝑘 𝑣 𝑘′)
(𝑣′
𝑘 𝑟 𝑘)
-
(( 𝑣 𝑘 𝑟′
𝑘 𝐻 𝑘)+ ( 𝐻 𝑘 𝑟𝑘 𝑣′
𝑘))
(𝑟′
𝑘 𝑣 𝑘)⁄
Família de Broyden
No método de Broyden utiliza-se uma correção dada por:
𝐶 𝑘 = (1 – α) 𝐶 𝑘
𝐷𝐹𝑃
+ α𝐶 𝑘
𝐵𝐹𝐺𝑆
No entanto, deve-se utilizar a fórmula para a estimativa da inversa da
Hessiana:
𝐻 𝑘+1 = 𝐻 𝑘 + 𝐶 𝑘 (α)
Enquanto α = 0 teremos o método de DFP e quando α = 1 teremos o método
BFGS.
2. Código Algoritmo Quase-Newton em Matlab
% Entradas:
% (1) A dimensionalidade dafunçãode múltiplasvariáveis.
% (2) Funçãono formatoadequado.
% OBS -> Nota:Por favor,garantirque a função inseridaestáno formatolegível peloMATLAB.
Este é o únicopontoem que há possibilidade de erromáximo.
% (3) vetorde aproximaçãoinicial.
% Nota:Garanta que a dimensionalidadecoincidecomdimensãode vetorde
aproximaçãoinicial.
% (4) Erro de tolerância
disp('Gostariade limparoseuespaçode trabalhode memóriae telade comando?');
z=input('Parafazerissodigite 1,qualqueroutronúmeroparacontinuar:');
if z==1
clc;clear
end
escolha=input('Parafazerumamaximizaçãoinserir1,para a minimizaçãodigite2:');a=2; %
coeficientea,dafunção;
while a==2
if escolha~=1&&escolha~=2
disp('Entradaerrada');a=2;
escolha=input('Parafazeramaximixaçãodigite 1,paraminimizardigite 2:');
else
a=1;
end
end
a=2;mark=0;flag=0;count=0;symsb;u=[];warningoff all
try
while a==2
4. disp('A dimensãodaaproximaçãoinicialé incorreto!');a=2;
x0=(input('Digiteaaproximaçãoinicial comovetorlinhade variaveis'));
else
a=1;
end
end
try
eps=abs(input('Digite umerrode tolerancia'));
catch
disp('Digite umnúmeroreal!');
end
g=f;
try
for i=1:n
h(i)=diff(g,['x'num2str(i)]);
forj=1:n
k(i,j)=diff(h(i),['x'num2str(j)]);
end
end
catch
disp('Seuproblemade otimizaçãonãopode serresolvido');
return
end
disp('MatrizHessianadafunção=')
if escolha==1
disp(-k)
elseif escolha==2
5. disp(k)
end
for i=1:n
for j=1:n
if i==j
d=det(-k(1:i,1:j));
if escolha==1
d=-d;
end
SOL=solve(d);
if str2num(char(d))<=0
mark=mark+1;u=[u i];
elseif isempty(SOL)==0
for m=1:length(SOL)
if isreal(SOL(m))==1||isa(SOL(m),'sym')
mark=mark+1;
if (isempty(find(u==i,1)))
u=[ui];
end
end
end
end
end
end
end
if mark>0
if escolha==1
6. fprintf('nO%gthmenorprincipal daHessiananãoé negativoparatoda reta real x!n',u);
fprintf('Entãoafunçãonãoé côncava globalmente,assimamaximizaçãonãoé
garantida!n');
elseif escolha==2
fprintf('nThe %gthmenorprincipal da Hessiananãoé positivoemtodaretareal x!n',u);
fprintf('Entãoafunção nãoé convexaglobalmente,e assimaminimizaçãoglobal nãoé
garantida!n');
end
else
if escolha==1
fprintf('nTodososmenoresprincipaisdaHessianasãonegativos,entãoafunçãoé
côncava globalmente!n');
disp('Assimamaximizaçãoglobal é possível!');
elseif escolha==2
fprintf('nTodososmenoresprincipaisdasHessianasãopositivos,assimafunçãoé
convexaglobalmente!n');
disp('Assimaminimizaçãoglobalé possível!');
end
end
X=x0;
if mark>0
S=eye(n);
else
S=k;
for i=1:n
S=(subs(S,['x'num2str(i)],X(i)));
end
end
7. %S=input('Digite umamatrizdefinidapositiva');
if n==2
A=[];B=[];C=f;
end
while flag~=1
count=count+1;steplength=0;
grad=h';fprintf('n-------------------------%gthIteração------------------nnn',count) % gth,
significan-ésimaposiçãoouposição;
if n==2
A=[A X(1)];B=[BX(2)];
end
for i=1:n
grad=subs(grad,['x'num2str(i)],X(i));
end
disp('pontoatual:');disp(X);
disp('Gradiente=');
if escolha==1
disp(-grad)
elseif escolha==2
disp(grad)
end
if max(abs(grad))>eps
fprintf('nA tolerânciade erroque você forneceunãofoi alcançadoaindan');
flag=input('Paraterminardigite1,senãooutronumeropara continuar:');
end
if flag==1
hes=k;
fori=1:n
8. hes=subs(hes,['x'num2str(i)],X(i));
end
fprintf('nn..........Resultado...........nn')
if length(find(eig(hes)>0))==length(eig(hes))
disp('Nopontoatual afunção é convexaentãoeladevaserum minimolocal !');
elseiflength(find(eig(hes)<0))==length(eig(hes))
disp('Nopontoatual afunção é concava assimeladevaseruma maximolocal');
else
disp('OPontoatual nãoé um EXTREMO local (pontocrítico)!');
end
disp('Atualmente,amatriz hessianaé:');
if escolha==1
disp(-hes)
elseifescolha==2
disp(hes)
end
fprintf('Opontoótimoem%gthoerroé:n',max(abs(grad)));
disp(X);
fori=1:n
f=subs(f,['x'num2str(i)],X(i));
end
fprintf('ne ovalordafunçãoaqui é:n');
if escolha==1
disp(-f)
elseifescolha==2
disp(f)
end
9. fprintf('Onúmerototal de iteraçõesrealizada=%gn',count);
end
if max(abs(grad))<=eps
fprintf('nnA tolerânciade erroque você forneceufoi alcançado.n');
hes=k;
fori=1:n
hes=subs(hes,['x'num2str(i)],X(i));
end
fprintf('nn..........Resultado...........nn')
if length(find(eig(hes)>0))==length(eig(hes))
disp('Alémdisso,nopresente momento,afunção é convexaporissopode serum
mínimolocal !');
elseiflength(find(eig(hes)<0))==length(eig(hes))
disp('Alémdisso,nopresente momento,afunçãoé côncava de modoque pode serum
máximolocal !');
else
disp('No entanto,opresentepontonãoé um pontoextremolocal !');
end
disp('Atualmente amatrizhesseanaé:');
if escolha==1
disp(-hes)
elseifescolha==2
disp(hes)
end
fprintf('Opontootimo em%gOerro é:n',max(abs(grad)));
disp(X);
fori=1:n
f=subs(f,['x'num2str(i)],X(i));
10. end
fprintf('nEaqui ovalorda funçãoé:n');
if escolha==1
disp(-f)
elseifescolha==2
disp(f)
end
fprintf('Onúmerototal de iteraçõesrealizada=%gn',count);flag=1;
elseif max(abs(grad))>eps&&flag~=1
fun=f;dir=-S*grad;
fori=1:n
fun=subs(fun,['x'num2str(i)],X(i)+b*dir(i));
end
diff(fun,b);
d=solve(diff(fun,b));
if isempty(d)==1
steplength=1;
else
t=double(d);
dd=diff(diff(fun,b));
fori=1:length(t)
if isreal(t(i))==1
if subs(dd,'b',t(i))>0
steplength=t(i);break
end
end
end
11. if steplength==0
for i=1:length(t)
if isreal(t(i))==1
if t(i)>0
steplength=t(i);break
end
end
end
end
if steplength==0
steplength=1;
end
end
funct=f;
fori=1:n
funct=subs(funct,['x'num2str(i)],X(i));
end
disp('Valorfuncional(dafunção) atualmente=');
if escolha==1
disp(-funct)
elseifescolha==2
disp(funct)
end
disp('Tamanhodopassotomado=');disp(steplength);
X=X+steplength*dir;p=steplength*dir;
grad_new=h';
fori=1:n
12. grad_new=subs(grad_new,['x'num2str(i)],X(i));
end
q=grad_new-grad;
S=S-(S*q*q'*S)/(q'*S*q)+(p*p')/(p'*q);
if count>100
disp('Você jádeve terfeito100iteraçõese parece que nenhumextremodafunção
existe!');
flag=input('Érecomendável paravocê encerraro processo,afazê-lo,digitando1,ou
qualqueroutronúmeroparacontinuar:');
end
end
end
try
if n==2&&length(A)>=2
[a,b]=meshgrid(A,B);C=subs(C,{x1,x2},{a,b});
view([-50,30]);axistight;holdon
surfc(a,b,C,'facecolor','green','edgecolor','b','facelighting','gouraud')
view([-50,30]);axistight;shadinginterp;
plot(A,B,'-mo',...
'LineWidth',2,...
'MarkerEdgeColor','k',...
'MarkerFaceColor',[.491 .63],...
'MarkerSize',12);
end
catch
disp('Nãofoi possível geraro"plot"de função(plotara função) !');
end