Constructing a convex function with prescribed Hessians at two given points

Question

Given two positive definite matrices $A, B \in \Bbb R^{n \times n}$, is there a way to construct a convex function $f: \Bbb R^n \to \Bbb R$ such that $$\nabla^2f(x)=A \qquad \text{and} \qquad \nabla^2f(y)=B$$ for two distinct $x,y \in \Bbb R^n$ (say $x=0_n$, $y=1_n$)?

I've tried the usual suspects (stuff with quadratics, exponentials, etc) without going anywhere.

Answer 1

Let us try to construct an example explicitly as $h(x)=f(g(x))$ where $f\colon\Bbb R\to\Bbb R$ and $g\colon\Bbb R^n\to\Bbb R$. The Hessian of $h$ is $$ \nabla^2 h(x)=f''(g(x))\nabla g(x)\nabla g(x)^T+f'(g(x))\nabla^2 g(x).\tag{1} $$ 1. Define for $T>0$ the function $$ f_T(t)=\begin{cases}\frac{t^2}{T}-\frac{t^3}{3T^2}, & t\le T\\ t-\frac{T}{3}, & t>T.\end{cases} $$ It is straightforward to verify that $f_T\in C^2$, convex and increasing for $t\ge 0$. Furthermore, $$ f_T'(0)=0,\quad f_T''(T)=0,\quad f_T'(T)=1.\tag{2} $$ 2. Define $g(x)=\frac12(x-a)^TB(x-a)$ for a fixed $a\in\Bbb R^n$ and a positive definite $B$. We have $$ \nabla g(x)=B(x-a),\quad\nabla^2 g(x)=B.\tag{3} $$ 3. Set $T=g(b)$ for a fixed $b\in\Bbb R^n$, $b\ne a$. Then $T>0$ and $h_1(x)=f_{g(b)}(g(x))$ is convex (as a composition of convex non-negative $g$ with convex and increasing for non-negative arguments $f$). Moreover, from (1)-(3) we have $$ \nabla^2 h_1(a)=f''(0)\cdot 0+0\cdot B=0,\quad \nabla^2 h_1(b)=0\cdot\nabla g(b)\nabla g(b)^T+1\cdot B=B. $$ 4. Similarly we can construct $h_2(x)$ with $$ \nabla^2 h_2(b)=0,\quad\nabla^2 h_2(a)=A $$ and take $h=h_1+h_2$.

Answer 2

Let $M$ be a positive number to be chosen later. For $z\in\mathbb{R}^n$, define $$f_1(z) = \frac12 \langle Az,z-x\rangle + M\langle z-x, x-y\rangle$$ and
$$f_2(z) = \frac12 \langle Bz,z-y\rangle + M\langle z-y, y-x\rangle$$
Note that $\nabla^2 f_1\equiv A$ and $\nabla^2 f_2\equiv B$; also, $f_1(x)=0$ and $f_2(y)=0$. Choose $M$ large enough so that $f_1(y)<0$ and $f_2(x)<0$; this is possible because the term multiplied by $M$ is negative in both cases.

Let $f=\max(f_1,f_2)$. This is a convex function whose Hessian is $A$ where $f_1>f_2$, and is $B$ where $f_1<f_2$. The inequalities $f_1>f_2$ and $f_1<f_2$ determine disjoint open sets $U,V$ that contain $x$ and $y$, respectively.

If a smooth function is desired, convolve $f$ with a compactly supported smooth bump $\phi$. When the diameter of $\operatorname{supp}\phi$ is smaller than $$\min(\operatorname{dist}(x, \partial U), \operatorname{dist}(y, \partial V))$$ the Hessian at $x$ will remain $A$ because in a neighborhood of $x$, $$\nabla^2 (f*\phi) = \nabla^2 (f_1*\phi) = (\nabla^2 f_1)*\phi=A$$ Same for $y$.