|
Quadratic Knapsack Problem
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of items to include in a collection without exceeding capacity of the knapsack, so as to maximize the overall profit. Usually, quadratic knapsack problems come with a restriction on the number of copies of each kind of item: either 0, or 1. This special type of QKP forms the 0-1 quadratic knapsack problem, which was first discussed by Gallo et al. The 0-1 quadratic knapsack problem is a variation of the knapsack problem, combining the features of the 0-1 knapsack problem and the quadratic knapsack problem. Definition Specifically, the 0–1 quadratic knapsack problem has the following form: : \text \left\ : \text X\equiv\left\. Here the binary variable ''xi'' represents whether ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Knapsack Problem
The knapsack problem is the following problem in combinatorial optimization: :''Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.'' It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The subset sum problem is a special case of the decision and 0-1 problems where each kind of item, the weight equals the value: w_i=v_i. In the field of cryptography, the term ''knapsack problem'' is often used to refer specifically to the subset sum pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linearization
In mathematics, linearization (British English: linearisation) is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization of a function Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that f(x) is differentiable on , b/math> (or , a/math>) and that a is close to b. In short, linearization approximates the output of a function near x = a. For example, \sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lagrange Multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variable (mathematics), variables). It is named after the mathematician Joseph-Louis Lagrange. Summary and rationale The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. In the general case, the Lagrangian is defined as \mathcal(x, \lambda) \equiv f(x) + \langle \lambda, g(x)\rangle for functions f, g; the notation \langle \cdot, \cdot \rangle denotes an inner prod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lagrangian Relaxation
In the field of mathematical optimization, Lagrangian relaxation is a relaxation method which approximates a difficult problem of constrained optimization by a simpler problem. A solution to the relaxed problem is an approximate solution to the original problem, and provides useful information. The method penalizes violations of inequality constraints using a Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the conditio ..., which imposes a cost on violations. These added costs are used instead of the strict inequality constraints in the optimization. In practice, this relaxed problem can often be solved more easily than the original problem. The problem of maximizing the Lagrangian function of the dual variables (the Lagrangian multipliers) is the Lagrangian dual problem. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Branch-and-bound
Branch and bound (BB, B&B, or BnB) is a method for solving optimization problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain the optimal solution. It is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores ''branches'' of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated ''bounds'' on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm. The algorithm depends on efficient estimation of the lower and u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Diagonally Dominant Matrix
In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is greater than or equal to the sum of the magnitudes of all the other (off-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if :, a_, \geq \sum_ , a_, \ \ \forall \ i where a_ denotes the entry in the ith row and jth column. This definition uses a weak inequality, and is therefore sometimes called ''weak diagonal dominance''. If a strict inequality (>) is used, this is called ''strict diagonal dominance''. The unqualified term ''diagonal dominance'' can mean both strict and weak diagonal dominance, depending on the context. Variations The definition in the first paragraph sums entries across each row. It is therefore sometimes called ''row diagonal dominance''. If one changes the definition to sum down each column, this is called ''column diagonal dominance''. Any strictly diagonally dominant matr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Positive Semi-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
