Lexicographic Max-min Optimization
Lexicographic max-min optimization (also called lexmaxmin or leximin or leximax or lexicographic max-ordering optimization) is a kind of multi-objective optimization. In general, multi-objective optimization deals with optimization problems with two or more objective functions to be optimized simultaneously. Lexmaxmin optimization presumes that the decision-maker would like the smallest objective value to be as high as possible; subject to this, the second-smallest objective should be as high as possible; and so on. In other words, the decision-maker ranks the possible solutions according to a leximin order of their objective function values. As an example, consider egalitarian social planners, who want to decide on a policy such that the utility of the poorest person will be as high as possible; subject to this, they want to maximize the utility of the second-poorest person; and so on. This planner solves a lexmaxmin problem, where the objective function number ''i'' is the utili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Multi-objective Optimization
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of MCDM, multiple-criteria decision making that is concerned with Mathematical optimization, mathematical optimization problems involving more than one Loss function, objective function to be optimized simultaneously. Multi-objective is a type of vector optimization that has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Interior Point Methods
Interior-point methods (also referred to as barrier methods or IPMs) are algorithms for solving linear and non-linear convex optimization problems. IPMs combine two advantages of previously-known algorithms: * Theoretically, their run-time is polynomial—in contrast to the simplex method, which has exponential run-time in the worst case. * Practically, they run as fast as the simplex method—in contrast to the ellipsoid method, which has polynomial run-time in theory but is very slow in practice. In contrast to the simplex method which traverses the ''boundary'' of the feasible region, and the ellipsoid method which bounds the feasible region from ''outside'', an IPM reaches a best solution by traversing the ''interior'' of the feasible region—hence the name. History An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programmin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
Constraint Satisfaction Problem
Constraint 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. CSPs often exhibit high complexity, requiring a combination of heuristics and combinatorial search methods to be solved in a reasonable time. Constraint programming (CP) is the field of research that specifically focuses on tackling these kinds of problems. Additionally, the Boolean satisfiability problem (SAT), satisfiability modulo theories (SMT), mixed integer programming (MIP) and answer set programming (ASP) are all fields of research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ordered Weighted Averaging Aggregation Operator
In applied mathematics, specifically in fuzzy logic, the ordered weighted averaging (OWA) operators provide a parameterized class of mean type aggregation operators. They were introduced by Ronald R. Yager.Yager, R. R., "On ordered weighted averaging aggregation operators in multi-criteria decision making," IEEE Transactions on Systems, Man, and Cybernetics 18, 183–190, 1988. Many notable mean operators such as the max, arithmetic average, median and min, are members of this class. They have been widely used in computational intelligence because of their ability to model linguistically expressed aggregation instructions. Definition An OWA operator of dimension \ n is a mapping F: \mathbb^n \rightarrow \mathbb that has an associated collection of weights \ W = _1, \ldots, w_n lying in the unit interval and summing to one and with : F(a_1, \ldots , a_n) = \sum_^n w_j b_j where b_j is the ''j''th largest of the a_i . By choosing different ''W'' one can implement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ronald R
Ronald is a masculine given name derived from the Old Norse ''Rögnvaldr'', Hanks; Hardcastle; Hodges (2006) p. 234; Hanks; Hodges (2003) § Ronald. or possibly from Old English '' Regenweald''. In some cases ''Ronald'' is an Anglicised form of the Gaelic '' Raghnall'', a name likewise derived from ''Rögnvaldr''. The latter name is composed of the Old Norse elements ''regin'' ("advice", "decision") and ''valdr'' ("ruler"). ''Ronald'' was originally used in England and Scotland, where Scandinavian influences were once substantial, although now the name is common throughout the English-speaking world. A short form of ''Ronald'' is ''Ron''. Pet forms of ''Ronald'' include ''Roni'' and '' Ronnie''. ''Ronalda'' and ''Rhonda'' are feminine forms of ''Ronald''. ''Rhona'', a modern name apparently only dating back to the late nineteenth century, may have originated as a feminine form of ''Ronald''. Hanks; Hardcastle; Hodges (2006) pp. 230, 408; Hanks; Hodges (2003) § Rhona. The names ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quasiconvex Function
In mathematics, a quasiconvex function is a real number, real-valued function (mathematics), function defined on an interval (mathematics), interval or on a convex set, convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex. ''Univariate'' Unimodality, unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple argument of a function, arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with Star_domain, star-convex sublevel sets can be unimodal without being quasiconvex. Def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographic Optimization
Lexicographic optimization is a kind of Multi-objective optimization. In general, multi-objective optimization deals with optimization problems with two or more objective functions to be optimized simultaneously. Often, the different objectives can be ranked in order of importance to the decision-maker, so that objective f_1 is the most important, objective f_2 is the next most important, and so on. Lexicographic optimization presumes that the decision-maker prefers even a very small increase in f_1, to even a very large increase in f_2, f_3, f_4, etc. Similarly, the decision-maker prefers even a very small increase in f_2, to even a very large increase in f_3, f_4, etc. In other words, the decision-maker has lexicographic preferences, ranking the possible solutions according to a lexicographic order of their objective function values. Lexicographic optimization is sometimes called preemptive optimization, since a small increase in one objective value preempts a much larger increas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Simplex Algorithm
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial ''cones'', and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function. History George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator. During 1946, his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, however, at tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dual Problem
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. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality. Dual problem Usually the term "dual problem" refers to the ''Lagrangian dual problem'' but o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ellipsoid Method
In mathematical optimization, the ellipsoid method is an iterative method for convex optimization, minimizing convex functions over convex sets. The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is polynomial in the input size. History The ellipsoid method has a long history. As an iterative method, a preliminary version was introduced by Naum Z. Shor. In 1972, an approximation algorithm for real convex optimization, convex minimization was studied by Arkadi Nemirovski and David B. Yudin (Judin). As an algorithm for solving linear programming problems with rational data, the ellipsoid algorithm was studied by Leonid Khachiyan; Khachiyan's achievement was to prove the Polynomial time, polynomial-time ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Leximin Order
In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division. Definition A vector x = (''x''1, ..., ''x''''n'') is ''leximin-larger'' than a vector y = (''y''1, ..., ''y''''n'') if one of the following holds: * The smallest element of x is larger than the smallest element of y; * The smallest elements of both vectors are equal, and the second-smallest element of x is larger than the second-smallest element of y; * ... * The ''k'' smallest elements of both vectors are equal, and the (''k''+1)-smallest element of x is larger than the (''k''+1)-smallest element of y. Examples The vector (3,5,3) is leximin-larger than (4,2,4), since the smallest element in the former is 3 and in the latter is 2. The vector (4,2,4) is leximin-larger than (5,3,2), since the smallest elements in both are 2, but the second-smallest elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |