Extra:Notation

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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: nth 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: ith element of vector LaTeX: x
  • LaTeX: e, e_i : vector of ones, LaTeX: ith unit vector
  • LaTeX: A_{(i,j}} : the element of matrix LaTeX: Ain the row LaTeX: i and column LaTeX: j
  • LaTeX: A_{(:,j)}, A_{(i,:)} : LaTeX: jth column of LaTeX: A, LaTeX: ith 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
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