Ask whether an optimum exists

Try

A synthetic workshop chooses quantities x and y of two products. Each quantity must be nonnegative, each uses time, and the workshop wants to maximize a stated profit. Before solving anything, write the units of every quantity and coefficient. Are partial products allowed? Is unused time permitted? Is profit linear for all quantities under consideration? These choices determine the feasible set and objective. Optimization finds the best decision within that mathematical model; it does not establish that the model faithfully describes the workshop.

Predict

Predict whether every problem with a finite best possible value has a decision attaining it. Minimize x on the open interval (0,1). Values can approach zero, but zero is not feasible. The infimum is zero and there is no minimum. Contrast minimizing x² on the whole real line: the feasible set is unbounded, yet the minimum is attained. Neither “bounded objective” nor “bounded feasible set” by itself is the complete existence criterion.

Experiment

Classify the following problems, distinguishing the optimal value from a point attaining it. For each missing optimum, identify a sequence of feasible points whose objective approaches the infimum.

Problem	Infimum	Attained?
Minimize x on [0,1]	0	Yes, at 0
Minimize x on (0,1)	0	No
Minimize e^(−x) on R	0	No
Minimize x² on R	0	Yes, at 0

The sequence x=1/n handles the second row, while x=n handles the third. These are different failures: a missing boundary point and escape to infinity. An algorithm returning a small number cannot distinguish attainment from nonattainment unless the mathematical structure is also examined.

Understand

A feasible point satisfies every constraint. A global minimizer has objective no larger than every feasible point; a local minimizer has that property only within some neighborhood intersected with the feasible set. Strictness means the inequality is strict for distinct feasible competitors in the relevant region. An infeasible problem has no feasible decisions. An unbounded-below minimization problem has infimum negative infinity. Nonattainment with finite infimum is a third, separate status.

The extreme-value theorem supplies existence when the feasible set is nonempty and compact and the objective continuous. One proof takes a minimizing sequence, extracts a convergent subsequence by compactness, and passes the objective values through continuity. In finite dimensions, closed and bounded sets are compact. If the feasible set is closed but unbounded, coercivity can substitute for boundedness: f(x) tends to positive infinity as the norm tends to infinity within that set. A nonempty sublevel set below one feasible objective value is then compact and contains a global minimizer.

Apply

For a concrete coercive example, minimize (x−2)²+(y+1)² subject to x+y≥4. The objective grows with distance from (2,−1), so large norms cannot improve indefinitely. The feasible half-plane is closed and nonempty. Existence follows before computing a solution. A projection calculation later identifies the closest feasible point as (7/2,1/2), with objective 9/2. Check feasibility and the squared distances independently; do not confuse proving existence with finding the optimizer.

Now change the constraint to x+y>4. The same boundary point is excluded. The infimum remains 9/2, approached by nearby points in the open half-plane, but is not attained. A strict inequality that looks like a small editing change can therefore alter the problem’s status.

Continue

Keep a model ledger containing decision variables, units, objective, feasible set, uncertainty and whether integrality is required. Record existence separately from a numerical candidate. Continue to Convex sets and projections to understand the geometry behind nearest-point certificates. Independent modeling and solver validation remain necessary even after a mathematically correct optimization theorem has been applied to the chosen formulation.