Reference the original problem text.

Now Suppose H(x,y) = 0 instead. (all other properties remain the same).

Can we can safely ignore H in the optimization and make progress towards the objective?

Question

Reference the original problem text.

Now Suppose H(x,y) >= 0 instead. (all other properties remain the same).

Can we can safely ignore H in the optimization and make progress towards the objective?

BlackTom AI · Accepted Answer

The question asks about the effect of assuming H(x,y) >= 0 and whether we can ignore H in the optimization. 
Option 1: True. If we suppose that H is always nonnegative, one might wonder if the H term can be dropped without affecting the progress toward the objective. However, this reasoning can be misleading depending on where H enters the problem. If H appears in the objective with a positive sign, then dropping it would artificially reduce the objective value and could yield an optimizer that is not truly optimal for the original problem. If H appears in a constraint, even a nonnegative H can influence feasibility or the feasible set in ways that affect the optimum. Therefore, claiming that we can safely ignore H simply because H is nonnegative overlooks how H interacts with the rest of the formulation, and this option risks an incorrect simplification. 
Option 2: False. In general, nonnegativity of H does not guarantee that H can be ignored in optimization. The presence of H could be essential either to the objective value or to feasibility, depending on whether H is part of the objective function or constraints. For example, if the objective is to minimize something like f(x,y) + H(x,y) with H ≥ 0, then removing H would lead to a different problem that may have a different optimum, potentially underestimating cost or mischaracterizing the feasible region. Similarly, if H is involved in a constraint such as g(x,y) ≤ H(x,y) or participates in a coupling constraint, the nonnegativity does not eliminate its influence on the feasible set. Consequently, you cannot just ignore H; you must assess its role in the model and how its presence or absence changes both feasibility and optimality. 
Option 1 (revisited): While nonnegativity can sometimes suggest a harmless addition, the correct caution is that ignoring H is not generally justified without a careful re-derivation of the problem. Relying solely on H ≥ 0 to justify removal overlooks cases where H contributes to the objective or to constraints in a way that alters the solution. 
Option 2 (revisited): The safe, broadly applicable stance is that you cannot assume ignore-it-and-progress is valid; you need to re-express the problem, verify whether H cancels out, and check if the reduced problem yields the same optimum as the original. This nuance is why the false statement is the appropriate general answer in many formulations.

CS-7643-O01, OAN, OSZ Quiz #5: Module 4

Reference the original problem text. Now Suppose H(x,y) >= 0 instead. (all other properties remain the same). Can we can safely ignore H in the optimization and make progress towards the objective?

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