Determine whether the constrained Newton solution is guaranteed to be a descent direction for a bound-constrained optimization problem | Step-by-Step Solution

What We're Solving:

We need to determine whether the constrained Newton solution provides a descent direction for a bound-constrained optimization problem. In other words, if we're minimizing a convex function f(x) subject to lower bounds (x ≥ l) and upper bounds (x ≤ u), will the constrained Newton step always point us "downhill" toward a lower function value?

The Approach:

This problem combines two key optimization concepts: Newton's method (which uses second-order information) and constrained optimization (where we can't move freely in all directions). We'll analyze this by understanding what makes a direction "descent" and how constraints modify the standard Newton direction.

Step-by-Step Solution:

Step 1: Understanding Descent Directions A direction d is a descent direction at point x if the directional derivative is negative: ∇f(x)ᵀd < 0. This means moving in direction d will initially decrease the function value.

Step 2: Review the Unconstrained Case For unconstrained problems, the Newton direction is d = -H⁻¹∇f(x), where H is the Hessian. When f is convex, H is positive semidefinite, making this a descent direction (assuming ∇f(x) ≠ 0).

Step 3: Understanding Constrained Newton Method For bound constraints, we modify the Newton step. At each variable xᵢ:

• If we're at a lower bound (xᵢ = lᵢ) and the gradient suggests moving lower, we "project" that component
• If we're at an upper bound (xᵢ = uᵢ) and the gradient suggests moving higher, we project that component
• Otherwise, we use the regular Newton direction for that component

Step 4: Analyzing the Descent Property The key insight: The constrained Newton direction is obtained by solving a quadratic program (QP) that minimizes the quadratic model while respecting bounds. This projection operation ensures we only move in feasible directions.

Step 5: The Critical Observation When we project the Newton direction onto the feasible region, we're essentially removing components that would violate constraints. Since the original Newton direction is descent for the unconstrained problem, and we're only removing components that would lead us away from the feasible region, the resulting direction maintains the descent property.

The Answer:

Yes, the constrained Newton solution IS guaranteed to be a descent direction for bound-constrained optimization of a convex function (assuming we're not already at the optimal solution).

The mathematical reasoning: The constrained Newton step solves: minimize ∇f(x)ᵀd + ½dᵀHd subject to feasible bounds

This quadratic program's solution will satisfy ∇f(x)ᵀd ≤ ∇f(x)ᵀd_Newton < 0, where d_Newton is the unconstrained Newton direction.

Memory Tip:

Think of it as "trimming the bad parts" - we start with a descent direction (Newton) and only remove components that would take us outside our allowed region. We never add components that make things worse, so we preserve the "downhill" property!

Remember: This beautiful result is why constrained Newton methods are so effective - they maintain the desirable convergence properties of Newton's method while respecting constraints.