Software Engineer · Phone Screen · Coding

Uber — Software Engineer ✅ Passed

Level: Unknown Level

Round: Phone Screen · Type: Coding · Difficulty: 6/10 · Duration: 60 min · Interviewer: Unfriendly

Topics: Binary Search, Math, Convex Function Minimization

Interview date: 2026-01-20

Summary

I was asked a codign question like this

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Finding the Minimum of a Convex Function via Black-Box Access

Problem Description

You are given access to a black-box function F(x) that is guaranteed to be convex over a closed interval ([a, b]). The internal definition of the function is unknown to you. The only way to interact with it is through a provided method:

evaluate(x)

which returns the value of F(x) for a given x.

Your goal is to identify a value x* within the interval ([a, b]) such that it is within a specified tolerance eps of the true minimizer of the function.

Formally, if x_min is the point where F(x) achieves its minimum, you must return an x* satisfying:

|x* - x_min| <= eps

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Key Observation

A convex function over a closed interval is unimodal, meaning it has exactly one global minimum. This structural property enables the use of efficient search techniques that progressively narrow the search range without evaluating every point.

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Constraints

• (-10^9 <= a < b <= 10^9)
• (0 <= eps <= 10^(-6))
• The function is convex on the entire interval ([a, b])
• You may only access the function via evaluate(x)

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Examples

Example 1

Input: Hidden function: (F(x) = (x - 3)^2 + 5) Interval: ([-10, 10]) Precision: eps = 0.01

Output: 3.0

Explanation: The function reaches its minimum at (x = 3). Any value in the range ([2.99, 3.01]) is considered a valid answer.

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Example 2

Input: Hidden function: (F(x) = x^2) Interval: ([-100, 50]) Precision: eps = 0.001

Output: 0.0

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Example 3

Input: Hidden function: (F(x) = |x - 123.456|) Interval: ([0, 1000]) Precision: eps = 0.0001

Output: 123.456

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Details

My approach:

`python import math

class ConvexFunction: def init(self, func): self.func = func

def evaluate(self, x: float) -> float:
    return self.func(x)

class Solution: def init(self, func: ConvexFunction): self.func = func

def minimize(self, a: float, b: float, eps: float) -> float:
    while (b - a) > eps:
        # Calculate two midpoints that divide the interval into three equal parts.
        m1 = a + (b - a) / 3
        m2 = b - (b - a) / 3

        f1 = self.func.evaluate(m1)
        f2 = self.func.evaluate(m2)

        if f1 < f2:
            b = m2
        else:
            a = m1
    return (a + b) / 2

def test1(): print("===== Test 1 =====") func = ConvexFunction(lambda x: (x - 3)**2 + 5) solution = Solution(func) result = solution.minimize(-10, 10, 0.01) print(f"Result: {result}") # Expected: ~3.0

def test2(): print("===== Test 2 =====") func = ConvexFunction(lambda x: x**2) solution = Solution(func) result = solution.minimize(-100, 50, 0.001) print(f"Result: {result}") # Expected: ~0.0

def test3(): print("===== Test 3 =====") func = ConvexFunction(lambda x: abs(x - 123.456)) solution = Solution(func) result = solution.minimize(0, 1000, 0.0001) print(f"Result: {result}") # Expected: ~123.456

def test4(): print("===== Test 4 =====") func = ConvexFunction(lambda x: (x - 987.654)**2) solution = Solution(func) result = solution.minimize(-1e9, 1e9, 1e-4) print(f"Result: {result}") # Expected: ~987.654

def test5(): print("===== Test 5 =====") func = ConvexFunction(lambda x: math.exp(x) - 2 * x) solution = Solution(func) result = solution.minimize(0, 2, 1e-4) print(f"Result: {result}") # Expected: ~0.693

def test6(): print("===== Test 6 =====") func = ConvexFunction(lambda x: x) solution = Solution(func) result = solution.minimize(0, 100, 1e-4) print(f"Result: {result}") # Expected: ~0.0

def test7(): print("===== Test 7 =====") func = ConvexFunction(lambda x: (x - 1)**2) solution = Solution(func) result = solution.minimize(0.9999, 1.0001, 1e-4) print(f"Result: {result}") # Expected: ~1.0

if name == 'main': test1() test2() test3() test4() test5() test6() test7()

`