Optimal Solution¶

Definition¶

For a minimization problem with feasible set $F$, a feasible vector $\boldsymbol{x}^*$ is an optimal solution if

For maximization the inequality is reversed. The number $z^*=f(\boldsymbol{x}^*)$ is the optimal value.

Interpretation¶

Optimality compares a feasible decision with every other allowed decision. An optimal solution is at least as good as each of them; it need not be strictly better, so several equally good decisions may exist.

For $z=3x_s+2x_t$ the dashed lines $z=60$ and $z=120$ cross the feasible region, so better plans exist. The line $z^*=180$ is the last one in the direction of increasing $z$ that still meets the region, and it touches only at $(20,60)$: this point is the optimal solution, and $180$ is the optimal value.

Both parts of the Definition matter: $\boldsymbol{x}^*$ must itself be feasible, and the inequality must hold for every $\boldsymbol{x}\in F$$F$, not only for the points tried so far. An optimum need not exist, since a model may have no feasible solution or an objective that improves without limit, and need not be unique: when an objective line lies along an edge of the region, every point of that edge is optimal.

Key Results¶

• A model may have no feasible solution or no finite optimum.
• The optimal value and the vector attaining it are different objects.
• An optimum is relative to the specified objective, constraints, and input data.

Examples¶

For the illustrated Giapetto objective, the optimal point $(20,60)$ has value $3\cdot20+2\cdot60=180$$180$.

• The manufacturing example has $x_1^*=2$$2$, $x_2^*=5$$5$, and $z^*=12$$12$.
• The practice LP with objective $2x_1+2x_2$ has optimum $(3,4)$ and value 14.

Prerequisites¶

• Feasible Solution
• Objective Function

Related Concepts¶

• Multiple Optimal Solutions
• Optimization Bounds

Sources¶

• OR Modelling · Week 1

Aliases: Optimum solution, optimal point Symbols: element of

Last updated: 20 October 2026 03:02