Given the sequence $a_1, a_2, \ldots, a_{100} \geq 0$ such that $\max\{a_{i-1} + a_i, a_i + a_{i+1}\} \geq i$ for any $2 \leq i \leq 99$, we need to find the minimum value of $a_1 + a_2 + \cdots + a_{100}$.

1. Analyzing the given condition:
$$\max\{a_{i-1} + a_i, a_i + a_{i+1}\} \geq i$$
This implies that either $a_{i-1} + a_i \geq i$ or $a_i + a_{i+1} \geq i$ (or both).

2. Constructing a sequence that satisfies the condition:
Let's consider a sequence where $a_i = \frac{i}{2}$ for $i = 1, 2, \ldots, 100$.

3. Verifying the sequence:
For $2 \leq i \leq 99$:
$$a_{i-1} + a_i = \frac{i-1}{2} + \frac{i}{2} = \frac{2i-1}{2}$$
$$a_i + a_{i+1} = \frac{i}{2} + \frac{i+1}{2} = \frac{2i+1}{2}$$
We need to check if:
$$\max\left\{\frac{2i-1}{2}, \frac{2i+1}{2}\right\} \geq i$$
Clearly, $\frac{2i+1}{2} \geq i$ for all $i$, since:
$$\frac{2i+1}{2} = i + \frac{1}{2} \geq i$$

4. Calculating the sum:
$$a_1 + a_2 + \cdots + a_{100} = \frac{1}{2} + \frac{2}{2} + \cdots + \frac{100}{2} = \frac{1 + 2 + \cdots + 100}{2}$$
Using the formula for the sum of the first $n$ natural numbers:
$$1 + 2 + \cdots + 100 = \frac{100 \cdot 101}{2} = 5050$$
Therefore:
$$a_1 + a_2 + \cdots + a_{100} = \frac{5050}{2} = 2525$$

5. Checking for a lower sum:
We need to check if there is a sequence that gives a lower sum while satisfying the condition. Consider the sequence where $a_i = 0$ for all $i$. This sequence does not satisfy the condition since $\max\{a_{i-1} + a_i, a_i + a_{i+1}\} = 0$ which is not $\geq i$.

Thus, the minimum sum of $a_1 + a_2 + \cdots + a_{100}$ that satisfies the given condition is $2525$.

The final answer is $\boxed{2525}$.