Lecture 2-1: Optimality Conditions

Lecture 2-1: Optimality Conditions#

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Global and Local Optima#

Local Minima#

Definition: Let \(f:S \to \mathbb{R}\), where \(S\subseteq \mathbb{R}^n\), then

  • \(x^*\) is a local minimum over \(S\) if \(\exists r > 0\) such that \(f(x) \geq f(x^*)\) \(\forall x \in S \cap B(x^*,r)\).

  • \(x^*\) is a strict local minimum over \(S\) if \(\exists r > 0\) such that \(f(x) > f(x^*)\) \(\forall x \in S \cap B(x^*,r)\).

Graph of a function on an interval showing a local minimum point marked as x*, where the function value is lower than all nearby points within a small neighborhood.

Local Maxima#

Definition: Let \(f:S \to \mathbb{R}\), where \(S\subseteq \mathbb{R}^n\), then

  • \(x^*\) is a local maximum over \(S\) if \(\exists r > 0\) such that \(f(x) \leq f(x^*)\) \(\forall x \in S \cap B(x^*,r)\).

  • \(x^*\) is a strict local maximum over \(S\) if \(\exists r > 0\) such that \(f(x) < f(x^*)\) \(\forall x \in S \cap B(x^*,r)\).

Graph of a function on an interval showing a local maximum point marked as x*, where the function value is higher than all nearby points within a small neighborhood.

Global Optima#

Definition: Let \(f:S \to \mathbb{R}\), where \(S\subseteq \mathbb{R}^n\), then \(x^* \in S\) is a

  • global minimum over \(S\) if \(f(x) \geq f(x^*)\) \(\forall x \in S\).

  • strict global minimum over \(S\) if \(f(x) > f(x^*)\) \(\forall x \in S\).

  • global maximum over \(S\) if \(f(x) \leq f(x^*)\) \(\forall x \in S\).

  • strict global maximum over \(S\) if \(f(x) < f(x^*)\) \(\forall x \in S\).

Graph of a function showing a global minimum point, where the function value is less than or equal to all other points in the domain.

Graph of a function showing a strict global minimum point, where the function value is strictly less than all other points in the domain.

Graph of a function showing a global maximum point, where the function value is greater than or equal to all other points in the domain.

Graph of a function showing a strict global maximum point, where the function value is strictly greater than all other points in the domain.

Generalized Extreme Value Theorem#

Theorem (Generalized Extreme Value Theorem): Let \(U \subset \mathbb{R}^n\) be a non-empty compact set. If \(f:U \to \mathbb{R}\) is a continuous function, then \(f\) is bounded and there exists \(p,q \in U\) such that \(f(p)=\sup_{x\in U}{f(x)}\) and \(f(q)=\inf_{x\in U}{f(x)}\).

Recall:

  • A subset of \(\mathbb{R}^n\) is compact if it is closed and bounded.

    • A set \(U\) is bounded if there exists \(M>0\) such that \(U\subset B(\mathbf{0},M)\), i.e., we can contain the set within a finite open ball.

    • A set \(U\) is closed with respect to a metric \(d\) if it contains all its limit points, i.e., \(U=\text{Cl}(U)\).

Critical Points#

Definition (Critical Points): Let \(f:U \to \mathbb{R}\). A point \(x^* \in U\) is called a critical point of \(f\) if \(\nabla f(x^*)=0\), or if \(\nabla f(x^*)\) is not defined.

A critical point can correspond to:

  • \(\nabla f(x^*)\) undefined (including boundary points of \(U\))

  • local minimum (\(\nabla f(x^*)=0\))

  • local maximum (\(\nabla f(x^*)=0\))

  • saddle point (\(\nabla f(x^*)=0\))

Graph of a function (such as an absolute value function) showing a critical point where the derivative is undefined, appearing as a sharp corner or cusp.

Graph of a function showing stationary points: a local maximum and a local minimum, both where the gradient equals zero and the tangent line is horizontal.

Three-dimensional surface plot of z = x^2 - y^2 showing a saddle point at the origin, which is a stationary point that is neither a local minimum nor a local maximum.

Stationary Points#

Definition (Stationary Points): Let \(f:U \to \mathbb{R}\), \(x^* \in \text{int}(U)\), and \(f\) be differentiable over a neighborhood of \(x^*\). Then \(x^*\) is called a stationary point of \(f\) if \(\nabla f(x^*)=0\).

A stationary point can correspond to:

  • local minimum (\(\nabla f(x^*)=0\))

  • local maximum (\(\nabla f(x^*)=0\))

  • saddle point (\(\nabla f(x^*)=0\))

Graph of a function showing stationary points: a local maximum and a local minimum, both where the gradient equals zero and the tangent line is horizontal.

Three-dimensional surface plot of z = x^2 - y^2 showing a saddle point at the origin, which is a stationary point that is neither a local minimum nor a local maximum.

Warning: Stationary points are critical points, but critical points are not necessarily stationary points.

First Order Optimality Condition#

Theorem (First Order Optimality Condition): Let \(f:U\to \mathbb{R}\) be a function defined on \(U\subseteq \mathbb{R}^n\). Suppose \(x^* \in \text{int}(U)\) is a local optimum (minimum or maximum) point and that all partial derivatives of \(f\) exist at \(x^*\).

Then \(\nabla f(x^*)=0\).

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