Showing that the infimum is a minimum

Question

Let $V > 0$ and let $\Phi(\cdot)$ be the standard normal CDF.

Consider the infimum of $$f(x_1, x_2,x_3, p_1, p_2, p_3) := p_1 \Phi(x_1) + p_2 \Phi(x_2) + p_3 \Phi(x_3)$$ with respect to $x_1, x_2, x_3, p_1, p_2$ and $p_3$ satisfying

$\begin{align} &p_1 \geq 0,\\\\ &p_2 \geq 0,\\\\ &p_3 \geq 0,\\\\ &p_1 + p_2 + p_3 = 1,\\\\ &p_1 x_1 + p_2 x_2 + p_3 x_3 = 0,\\\\ &p_1 x_1^2 + p_2 x_2^2 + p_3 x_3^2 \leq V. \end{align} $

The objective is a continuous function and the feasible set is a subset of $\mathbb{R}^6$. One way to show that the infimum is a minimum is to show that the feasible set is compact. But even with the constraint $p_1 x_1^2 + p_2 x_2^2 + p_3 x_3^2 \leq V$, it can happen that $p_1$ is arbitrarily small and $x_1$ is arbitrarily large.

I think I need to use the fact that the objective function is bounded between $0$ and $1$, which together with the last constraint limits $x_i$'s becoming very large.

Answer 1

$\newcommand{\R}{\mathbb R}\newcommand{\tx}{\tilde x}$Indeed, the infimum (say $f^*$) in question is attained for each $V\in(0,\infty)$.

To prove this, for each $j\in\{1,2,3\}$ let $(x_j^n)_{n=1}^\infty$ and $(p_j^n)_{n=1}^\infty$ be sequences such that for each $n$ all your conditions are satisfied with $x_1^n,x_2^n,x_3^n,p_1^n,p_2^n,p_3^n$ in place of $x_1,x_2,x_3,p_1,p_2,p_3$ and \begin{equation} f(x_1^n,x_2^n,x_3^n,p_1^n,p_2^n,p_3^n)\to f^* \end{equation} (as $n\to\infty$). Passing to subsequences, without loss of generality assume that for each $j\in\{1,2,3\}$ \begin{equation} x_j^n\to x_j^*\in[-\infty,\infty],\quad p_j^n\to p_j^*\in[0,1]. \end{equation} Let \begin{equation} J:=\{j\in\{1,2,3\}\colon x_j^*\in\R\},\quad J^c:=\{1,2,3\}\setminus J. \end{equation} Since $p_j^n (x_j^n)^2\le V<\infty$, we see that for each $j\in J^c$ we have $p_j^*=0$ and $p_j^n x_j^n\to0$. So, letting $\tx_j:=x_j^*\,1(j\in J)$ for $j\in\{1,2,3\}$, we have \begin{equation} p_j^*\ge0\ \forall j\in\{1,2,3\},\quad \sum_{j=1}^3 p_j^*=1, \end{equation} \begin{equation} \sum_{j=1}^3 p_j^*\tx_j=\sum_{j\in J} p_j^*x_j^*=\lim_n \sum_{j=1}^3 p_j^n x_j^n=\lim_n 0=0, \end{equation} \begin{equation} \sum_{j=1}^3 p_j^*\tx_j^2=\sum_{j\in J} p_j^*(x_j^*)^2=\lim_n \sum_{j\in J}p_j^n (x_j^n)^2 \le\limsup_n \sum_{j=1}^3 p_j^n (x_j^n)^2\le V, \end{equation} \begin{equation} \sum_{j=1}^3 p_j^*\Phi(\tx_j)=\sum_{j\in J} p_j^*\Phi(x_j^*) =\lim_n \sum_{j\in J} p_j^n \Phi(x_j^n) =\lim_n \sum_{j=1}^3 p_j^n \Phi(x_j^n)=f^*. \end{equation}

So, all your conditions are satisfied with $\tx_1,\tx_2,\tx_3,p_1^*,p_2^*,p_3^*$ in place of $x_1,x_2,x_3,p_1,p_2,p_3$ and the infimum $f^*$ is attained at $(x_1,x_2,x_3,p_1,p_2,p_3)=(\tx_1,\tx_2,\tx_3,p_1^*,p_2^*,p_3^*)$. $\quad\Box$