# Notation

General

• $LaTeX: C^n$: class of functions with $LaTeX: n$ continuous derivitives.
• $LaTeX: \textstyle \nabla f(x)$: gradient of $LaTeX: f$ at $LaTeX: x$
• $LaTeX: \textstyle \nabla_x f(x,y)$: gradient of $LaTeX: f$ at $LaTeX: (x,y)$
• $LaTeX: H_f(x)$: Hessian of $LaTeX: f$ at $LaTeX: x$
• $LaTeX: \textstyle \sum_{i \in I} x_i$, $LaTeX: \textstyle \sum_{i=1}^n x_i$, $LaTeX: \textstyle \sum_{i} x_i$ : sum of $LaTeX: x_i$ over an appropriate index
• $LaTeX: |x|$ : absolute value
• $LaTeX: \lfloor x \rfloor$ : integer part, or floor, of $LaTeX: x$
• $LaTeX: x \vee y$ : the join of $LaTeX: x$ and $LaTeX: y$
• $LaTeX: x \wedge y$ : the meet of $LaTeX: x$ and $LaTeX: y$
• $LaTeX: x^- = \min\{0, x\}$ : negative part of $LaTeX: x$
• $LaTeX: x^+ = \max\{0, x\}$ : positive part of $LaTeX: x$
• $LaTeX: \| x \|, \| x \|_p$: ($LaTeX: p$-)norm of $LaTeX: x$
• $LaTeX: \textstyle O(f) = \{ g : |g(x)| \le M |f(x)| \mbox{ for some M independent of x}\}$ : class of functions whose value is bounded by a uniform multiple of the magnitude of $LaTeX: f$, where the domain of $LaTeX: x$ is selected appropriately
• $LaTeX: \sqrt[n]{x}$ : $LaTeX: n$th root of $LaTeX: x$

Sets and Set Operations

• $LaTeX: \sim S$ : complement of $LaTeX: S$
• $LaTeX: S\backslash T$ : $LaTeX: S$ without $LaTeX: T$
• $LaTeX: S \cap T, \cap_{i \in I} S_i, \cap_i S_i, \cap S_i$ : intersection of $LaTeX: S$ and $LaTeX: T$ or of a sequence of sets $LaTeX: S_i$
• $LaTeX: S \times T, \textstyle\prod_{i \in I} S_i, \prod_i S_i, \prod S_i$ : (cross) product of $LaTeX: S$ with $LaTeX: T$ or of a sequence of sets $LaTeX: S_i$
• $LaTeX: S \cup T, \cup_{i \in I} S_i, \cup_i S_i, \cup S_i$ : union of $LaTeX: S$ and $LaTeX: T$ or of a sequence of sets $LaTeX: S_i$
• $LaTeX: \mbox{cl}(S)$ : closure of $LaTeX: S$
• $LaTeX: \mbox{ext}(S)$ : extreme points of $LaTeX: S$
• $LaTeX: \mbox{int}(S)$ : interior of $LaTeX: S$
• $LaTeX: \mbox{ri}(S)$ : relative interior of $LaTeX: S$
• $LaTeX: \arg\min\{f(x) : x \in X\} = \{x^* \in X : f(x^*) \le f(x) \mbox{ for all } x \in X \}$ : argument minimum of $LaTeX: f$ over $LaTeX: X$
• $LaTeX: \arg\max\{f(x) : x \in X\} = \{x^* \in X : f(x^*) \ge f(x) \mbox{ for all } x \in X \}$ : argument maximum of $LaTeX: f$ over $LaTeX: X$
• $LaTeX: \mbox{dom}(f)$ : effective domain of $LaTeX: f$
• $LaTeX: \emptyset$ : empty set
• $LaTeX: 2^S$ : power set, all subsets, of $LaTeX: S$
• $LaTeX: \mathbb{R}^n$ : reals cross itself $LaTeX: n$ times -i.e. real $LaTeX: n$-vectors
• $LaTeX: \mathbb{R}^n_+, \mathbb{R}^n_{++}$ : componentwise non-negative and positive elements of $LaTeX: \mathbb{R}^n$
• $LaTeX: \mathbb{Z}^n$ : integers cross itself $LaTeX: n$ times -i.e. integer $LaTeX: n$-vectors
• $LaTeX: \mathbb{Z}^n_+, \mathbb{Z}^n_{++}$ : componentwise non-negative and positive elements of $LaTeX: \mathbb{Z}^n$
• $LaTeX: [a, b]$ : closed line segment
• $LaTeX: (a, b)$ : open line segment
• $LaTeX: S_n$ : simplex in $LaTeX: \mathbb{R}^n$

Vectors and Matrices

• $LaTeX: x_i$ : $LaTeX: i$th element of vector $LaTeX: x$
• $LaTeX: e, e_i$ : vector of ones, $LaTeX: i$th unit vector
• $LaTeX: A_{(i,j}}$ : the element of matrix $LaTeX: A$in the row $LaTeX: i$ and column $LaTeX: j$
• $LaTeX: A_{(:,j)}, A_{(i,:)}$ : $LaTeX: j$th column of $LaTeX: A$, $LaTeX: i$th row of $LaTeX: A$
• $LaTeX: A^{-1}$ : inverse of $LaTeX: A$
• $LaTeX: A^T$ : transpose of $LaTeX: A$
• $LaTeX: \mbox{diag}(x)$ : diagonal matrix whose main diagonal is the vector $LaTeX: x$
• $LaTeX: I$ : identity matrix

Abbreviations

 AI artificial intelligence ANN artificial neural network B&B branch and bound BFGS Broyden-Fletcher-Goldfarb-Shanno BLP Bilevel program BTRAN backward transformation CP complementarity problem DFP Davidson-Fletcher-Powell DP dynamic program FIFO first in, first out (pertains to a list processing rule) FTRAN forward transformation GA genetic algorithm GLM Generalized Lagrange Multiplier method GRASP Greedy Randomized Adaptive Search Procedures GRG generalized reduced gradient method GUB generalized upper bound IIS irreducible inconsistent subsystem IP integer program IPM interior point method LCP linear complementarity problem LMR Lagrange Multiplier Rule LP linear program LIFO last in, first out (pertains to a list processing rule) MILP mixed integer linear program MINLP mixed integer nonlinear program MIP mixed integer program MIQP mixed integer quadratic program MOLP multiple objective linear program MOMP multiple objective mathematical program MPEC Mathematical Program with Equilibrium Constraints MPS Mathematical Programming System NLP nonlinear program OBJ objective function OR Operations Research PARTAN parallel tangents QAP quadratic assignment problem QP quadratic program QQCP quadratic program with quadratic constraints RHS right-hand side RO robust optimization SLP sequential linear programming SOS specially ordered set SP stochastic program SQP sequential quadratic programming SUMT sequential unconstrained minimization technique TSP traveling salesman problem VRP vehicle routing problem VUB variable upper bound