Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ and each $a_i$ are fixed and distinct and positive and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always produces a vertex point on the simplex and maximization corresponds to $\max_i a_i$.

1. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at maximizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

Suppose we are looking for a $0/1$ vector in integers (not reals as in 1.) on the standard simplex and we know the optimal vector has either sum of even coordinates summing to $1$ or odd coordinates summing to $1$ then in finite dimensions it is a process of enumerating vertices.

2. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at optimizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

In general is the proofs of optimization over infinite dimensions considered constructive?

Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ and each $a_i$ are fixed and non-negative and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always produces a vertex point on the simplex and maximization corresponds to $\max_i a_i$.

1. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at maximizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

Suppose we are looking for a $0/1$ vector in integers (not reals as in 1.) on the standard simplex and we know the optimal vector has either sum of even coordinates summing to $1$ or odd coordinates summing to $1$ then in finite dimensions it is a process of enumerating vertices.

2. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at optimizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

In general is the proofs of optimization over infinite dimensions considered constructive?

3a. How about when each $a_i$ are fixed and positive?

3b. How about when each $a_i$ are fixed and distinct and non-negative thus guaranteeing an unique vertex point?

3c. How about when each $a_i$ are fixed and distinct and positive thus guaranteeing an unique vertex point?

Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ are fixed and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always produces a vertex point on the simplex and maximization corresponds to $\max_i a_i$.

1. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at maximizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

Suppose we are looking for a $0/1$ vector in integers (not reals as in 1.) on the standard simplex and we know the optimal vector has either sum of even coordinates summing to $1$ or odd coordinates summing to $1$ then in finite dimensions it is a process of enumerating vertices.

2. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at optimizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

In general is the proofs of optimization over infinite dimensions considered constructive?

Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ and each $a_i$ are fixed and distinct and positive and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always produces a vertex point on the simplex and maximization corresponds to $\max_i a_i$.

1. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at maximizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

Suppose we are looking for a $0/1$ vector in integers (not reals as in 1.) on the standard simplex and we know the optimal vector has either sum of even coordinates summing to $1$ or odd coordinates summing to $1$ then in finite dimensions it is a process of enumerating vertices.

2. In infinite dimensions is such a proof considered constructive or does it hold only in classical logic? It seems that we would have to show that at optimizing point there is an $i\in\mathbb N$ such that $x_i=1$ holds and for that perhaps is it possible we cannot do this without invoking LLPO?

In general is the proofs of optimization over infinite dimensions considered constructive?