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{\bf Math 273}: {\bf Homework \#1, due on Monday, October 11}
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{\bf 1.} Compute the gradient $\nabla f(x)$ and Hessian $\nabla^2 f(x)$ of the function
$$f(x)=100(x_2-x_1^2)^2+(1-x_1)^2.$$
Show that $x^*=(1,1)^T$ is the only local minimizer of this function, and that the Hessian matrix at that point is positive definite.
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{\bf 2.} Let $a$ be a given $n$-vector, and $A$ be a given $n\times n$ symmetric matrix. Compute the gradient and Hessian of $f_1(x)=a^Tx$ and $f_2(x)=x^TAx$.
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{\bf 3.} Suppose that $f$ is a convex function. Show that the set of global minimizers of $f$ is a convex set.
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{\bf 4.} Suppose that $\hat f(z)=f(x)$, where $x=Sz+s$ for some $S\in R^{n\times n}$ and $s\in R^n$. Show that
$$\nabla \hat f(z)=S^T\nabla f(x),\ \ \ \ \nabla ^2\hat f(z)=S^T\nabla^2 f(x)S.$$
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%{\bf 5.} (Directional derivative) Let $f:R^n\rightarrow R$ be continuously differentiable. Show that
%$$\lim_{\epsilon\rightarrow0}\frac{f(x+\epsilon p)-f(x)}{\epsilon}=\nabla f(x)^Tp.$$
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{\bf 5.} {\it Computation of the Euler-Lagrange equation in the continuous case}
Consider the minimization problem in two dimensions $(x,y)$,
$$\inf_{u}E(u)=\int_{\Omega}L(x,y,u,u_x,u_y)dxdy,\ \ \ u=g\mbox{ on }\partial\Omega,$$
where $g$ is a given function on the boundary $\partial\Omega$, with $\Omega$ an bounded and bounded region in the plane. Assume that the integrand $L$ is differentiable.
(i) Show that a sufficiently smooth minimizer $u$ formally satisfies the Euler-Lagrange equation
$$\frac{\partial}{\partial x} L_{u_x}(P)+\frac{\partial}{\partial y}L_{u_y}(P)-L_u(P)=0$$
on $\Omega$, where $P=(x,y,u(x,y),u_x(x,y),u_y(x,y))$.
(ii) Apply the above result to the case when $L(x,y,u_x,u_y)=u_x^2+u_y^2-2fu$.
Hint: consider another test function $v$, such that $v=0$ on $\partial \Omega$. Since $u$ is a minimizer, we must have $E(u)\leq E(u+\varepsilon v)$ for all such sufficiently smooth functions $v$ and all real $\epsilon$. Apply integration by parts to obtain the desired result. Here, $(u_x,u_y)=\nabla u$.
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