Duality (optimization)In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem.
Mathematical optimizationMathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
Linear programming relaxationIn mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form The relaxation of the original integer program instead uses a collection of linear constraints The resulting relaxation is a linear program, hence the name.
Primality testA primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input).
Integer programmingAn integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.
Convex optimizationConvex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.
Combinatorial optimizationCombinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.
Primality certificateIn mathematics and computer science, a primality certificate or primality proof is a succinct, formal proof that a number is prime. Primality certificates allow the primality of a number to be rapidly checked without having to run an expensive or unreliable primality test. "Succinct" usually means that the proof should be at most polynomially larger than the number of digits in the number itself (for example, if the number has b bits, the proof might contain roughly b2 bits).
Verdier dualityIn mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism.
Duality (mathematics)In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings.
Poincaré dualityIn mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the ()th homology group of M, for all integers k Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
Optimization problemIn mathematics, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
Constraint satisfaction problemConstraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables, which is solved by constraint satisfaction methods. CSPs are the subject of research in both artificial intelligence and operations research, since the regularity in their formulation provides a common basis to analyze and solve problems of many seemingly unrelated families.
Elliptic curve primalityIn mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and de, in 1993. The concept of using elliptic curves in factorization had been developed by H. W.
Optimal controlOptimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.
AlpsThe Alps (ælps) are the highest and most extensive mountain range that is entirely in Europe, stretching approximately across eight Alpine countries (from west to east): Monaco, France, Switzerland, Italy, Liechtenstein, Germany, Slovenia, and Austria. The Alpine arch extends from Nice on the western Mediterranean to Trieste on the Adriatic and Vienna at the beginning of the Pannonian Basin. The mountains were formed over tens of millions of years as the African and Eurasian tectonic plates collided.
Swiss AlpsThe Alpine region of Switzerland, conventionally referred to as the Swiss Alps (Schweizer Alpen, Alpes suisses, Alpi svizzere, Alps svizras), represents a major natural feature of the country and is, along with the Swiss Plateau and the Swiss portion of the Jura Mountains, one of its three main physiographic regions. The Swiss Alps extend over both the Western Alps and the Eastern Alps, encompassing an area sometimes called Central Alps.
Eastern AlpsEastern Alps is the name given to the eastern half of the Alps, usually defined as the area east of a line from Lake Constance and the Alpine Rhine valley up to the Splügen Pass at the Alpine divide and down the Liro River to Lake Como in the south. The peaks and mountain passes are lower than the Western Alps, while the range itself is broader and less arched.
Bachelor of Fine ArtsA Bachelor of Fine Arts (BFA) is a standard undergraduate degree for students for pursuing a professional education in the visual, fine or performing arts. It is also called Bachelor of Visual Arts (BVA) in some cases. The Bachelor of Fine Arts degree differs from a Bachelor of Arts (BA) degree in that the majority of the program consists of a practical studio component, as contrasted with lecture and discussion classes.
Fine artIn European academic traditions, fine art is made primarily for aesthetics or creative expression, distinguishing it from decorative art or applied art, which also has to serve some practical function, such as pottery or most metalwork. In the aesthetic theories developed in the Italian Renaissance, the highest art was that which allowed the full expression and display of the artist's imagination, unrestricted by any of the practical considerations involved in, say, making and decorating a teapot.
