 # Unbounded math program

The objective is not bounded on the feasible region  (from above, if maximizing; from below, if minimizing). Equivalently, there exists a sequence of feasible points, say $LaTeX: \textstyle x^k$ for which $LaTeX: \textstyle f(x^k)$ diverges to infinity (minus infinity if minimizing).

# Unconstrained math program

One with no constraints (can still have $LaTeX: X$ be a proper subset of $LaTeX: \textstyle \mathbb{R}^n,$ such as requiring $LaTeX: x$ to be integer-valued.)

# Unconstrained optimization

Taken literally, this is an unconstrained mathematical program. However, this phrase is also used in a context that $LaTeX: X$ could contain the strict interior, with constraints of the form $LaTeX: g(x) < 0,$ but the mathematical program behaves as unconstrained. This arises in the context of some algorithm design, as the solution is known to lie in the interior of $LaTeX: X,$ such as with the barrier function. As long as the algorithm has a continuous trajectory, this abuse is ok, but the constraints must be considered if one could jump, as in pattern search.

# Unimodal function

Has one mode (usually a maximum, but could mean a minimum, depending on context). If f is defined on the interval $LaTeX: [a, b],$ let $LaTeX: x^*$ be its mode. Then, $LaTeX: f$ strictly increases from $LaTeX: a$ to $LaTeX: x^*$ and strictly decreases from $LaTeX: x^*$ to $LaTeX: b$ (reverse the monotonicity on each side of $LaTeX: x^*$ if the mode is a minimum). (For line search methods, like fibonacci, the mode could occur in an interval, $LaTeX: [a^*,b^*],$ where $LaTeX: f$ strictly increases from $LaTeX: a$ to $LaTeX: a^*,$ is constant (at its global max value) on $LaTeX: [a^*,b^*],$ then strictly decreases on $LaTeX: [b^*,b].$)

# Unimodular matrix

A nonsingular matrix whose determinate has magnitude 1. A square matrix is totally unimodular if every nonsingular submatrix from it is unimodular. This arises in (linear) integer programming because it implies a basic solution to the LP relaxation is integer-valued (given integer-valued right-hand sides), thus obtaining a solution simply by a simplex method. An example of a totally unimodular matrix is the node-arc incidence matrix of a network, so basic solutions of network flows are integer-valued (given integer-valued supplies and demands).

# Unitary matrix

A nonsingular matrix whose Hermitian adjoint equals its inverse (same as orthogonal for real-valued matrices).

# Univariate optimization

A mathematical program with a single variable.

# Upper semi-continuity

(or upper semi-continuous, abbr. usc). Suppose $LaTeX: \textstyle \{x^k\} \to x.$ Of a function, $LaTeX: \textstyle \limsup f(x^k) \le f(x).$ Of a point-to-set map, let $LaTeX: \textstyle \mbox{N}_e\lbrack\mbox{S}\rbrack$ be a neighborhood of the set S. For each e > 0, there exists K such that for all $LaTeX: \textstyle k > \mbox{K}, \mbox{A}(x^k)$ is a subset of $LaTeX: \textstyle \mbox{N}_e\lbrack\mbox{A}(x)\rbrack.$ Here is an example of what can go wrong. Consider the feasibility map with $LaTeX: g(x) = \begin{cases}

(x+\sqrt{2})^2 - 1, & \mbox{if } x < 0 \\ e^{-x}, & \mbox{if } x \ge 0

\end{cases}$

Note g is continuous and its level set is $LaTeX: \textstyle \lbrack -\sqrt{2} - 1, -\sqrt{2} + 1 \rbrack.$ However, for any $LaTeX: \textstyle b > 0, \{x: g(x) \le b\} = \lbrack - \sqrt{2} - 1 - b, - \sqrt{2} + 1 + b \rbrack \lor \lbrack - \log b, \inf),$

which is not bounded. The map fails to be usc at 0 due to the lack of stability of its feasibility region when perturbing its right-hand side (from above).

# Upper triangular matrix

A square matrix, $LaTeX: A,$ such that $LaTeX: \textstyle \mbox{A}_{i, j} = 0 \mbox{ for } i \ge j.$

# Utility function

A measure of benefit, used as a maximand in economic models.