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Welfare Maximization Problem
The welfare maximization problem is an optimization problem studied in economics and computer science. Its goal is to partition a set of items among agents with different utility functions, such that the welfare – defined as the sum of the agents' utilities – is as high as possible. In other words, the goal is to find an item allocation satisfying the utilitarian rule. An equivalent problem in the context of combinatorial auctions is called the winner determination problem. In this context, each agent submits a list of bids on sets of items, and the goal is to determine what bid or bids should win, such that the sum of the winning bids is maximum. Definitions There is a set ''M'' of ''m'' items, and a set ''N'' of ''n'' agents. Each agent ''i'' in ''N'' has a utility function u_i: 2^M \to \mathbb. The function assigns a real value to every possible subset of items. It is usually assumed that the utility functions are monotone set functions, that is, Z_1\supseteq Z_2 implies u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Problem
In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ..., 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. * A problem with continuous variables is known as a '' continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Search space In the context of an optim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Matroid
In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a ''capacity constraint'' - a maximum number of allowed elements from this category. The independent sets of a partition matroid are exactly the sets in which, for each category, the number of elements from this category is at most the category capacity. Formal definition Let C_i be a collection of disjoint sets ("categories"). Let d_i be integers with 0\le d_i\le , C_i, ("capacities"). Define a subset I\subseteq \bigcup_i C_i to be "independent" when, for every index i, , I\cap C_i, \le d_i. The sets satisfying this condition form the independent sets of a matroid, called a partition matroid. The sets C_i are called the categories or the blocks of the partition matroid. A basis of the partition matroid is a set whose intersection with every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demand Oracle
In algorithmic game theory, a branch of both computer science and economics, a demand oracle is a function that, given a price-vector, returns the demand of an agent. It is used by many algorithms related to pricing and optimization in online market. It is usually contrasted with a value oracle, which is a function that, given a set of items, returns the value assigned to them by an agent. Demand The demand of an agent is the bundle of items that the agent most prefers, given some fixed prices of the items. As an example, consider a market with three objects and one agent, with the following values and prices. Suppose the agent's utility function is additive (= the value of a bundle is the sum of values of the items in the bundle), and quasilinear (= the utility of a bundle is the value of the bundle minus its price). Then, the demand of the agent, given the prices, is the set , which gives a utility of (4+6)-(3+1) = 6. Every other set gives the agent a smaller utility. For exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Sampling
In this statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. The subset is meant to reflect the whole population, and statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population (in many cases, collecting the whole population is impossible, like getting sizes of all stars in the universe), and thus, it can provide insights in cases where it is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a trajectory that maximizes that function; the procedure is then known as ''gradient ascent''. It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
With High Probability
In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number ''n'' and goes to 1 as ''n'' goes to infinity, i.e. the probability of the event occurring can be made as close to 1 as desired by making ''n'' big enough. Applications The term WHP is especially used in computer science, in the analysis of probabilistic algorithms. For example, consider a certain probabilistic algorithm on a graph with ''n'' nodes. If the probability that the algorithm returns the correct answer is 1-1/n, then when the number of nodes is very large, the algorithm is correct with a probability that is very near 1. This fact is expressed shortly by saying that the algorithm is correct WHP. Some examples where this term is used are: * Miller–Rabin primality test: a probabilistic algorithm for testing whether a given number ''n'' is prime or composite. If ''n'' is composite, the test will detect ''n'' as composite WHP. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Vondrák
Jan Vondrák is a Czech applied mathematician and theoretical computer scientist. He has been a professor of mathematics at Stanford University since 2015. He was a research staff member in the theory group at the IBM Almaden Research Center from 2009 to 2015. Vondrák completed a bachelor's degree in physics (1995) and an M.S. (1999) and Ph.D. (2007) in computer science at Charles University under advisor Martin Loebl. He met mathematician Maryam Mirzakhani in 2004 in Boston. Vondrák completed a Ph.D. in applied mathematics in 2005 at Massachusetts Institute of Technology under advisor Michel Goemans. He was a postdoctoral researcher in the theory group at Microsoft Research from 2005 to 2006. From 2006 to 2009, Vondrák was a postdoctoral teaching fellow at Princeton University. He married Mirzakhani in 2008 on a mountain in New Hampshire New Hampshire ( ) is a U.S. state, state in the New England region of the Northeastern United States. It borders Massachusetts to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Utility
Marginal utility, in mainstream economics, describes the change in ''utility'' (pleasure or satisfaction resulting from the consumption) of one unit of a good or service. Marginal utility can be positive, negative, or zero. Negative marginal utility implies that every consumed additional unit of a commodity causes more harm than good, leading to a decrease in overall utility. In contrast, positive marginal utility indicates that every additional unit consumed increases overall utility. In the context of cardinal utility, liberal economists postulate a law of diminishing marginal utility. This law states that the first unit of consumption of a good or service yields more satisfaction or utility than the subsequent units, and there is a continuing reduction in satisfaction or utility for greater amounts. As consumption increases, the additional satisfaction or utility gained from each additional unit consumed falls, a concept known as ''diminishing marginal utility.'' This idea is us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Value Oracle
In algorithmic game theory, a branch of both computer science and economics, a demand oracle is a function that, given a price-vector, returns the demand of an agent. It is used by many algorithms related to pricing and optimization in online market. It is usually contrasted with a value oracle, which is a function that, given a set of items, returns the value assigned to them by an agent. Demand The demand of an agent is the bundle of items that the agent most prefers, given some fixed prices of the items. As an example, consider a market with three objects and one agent, with the following values and prices. Suppose the agent's utility function is additive (= the value of a bundle is the sum of values of the items in the bundle), and quasilinear (= the utility of a bundle is the value of the bundle minus its price). Then, the demand of the agent, given the prices, is the set , which gives a utility of (4+6)-(3+1) = 6. Every other set gives the agent a smaller utility. For exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decreasing Marginal Return
In economics, diminishing returns means the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal (''ceteris paribus''). The law of diminishing returns (also known as the law of diminishing marginal productivity) states that in a productive process, if a factor of production continues to increase, while holding all other production factors constant, at some point a further incremental unit of input will return a lower amount of output. The law of diminishing returns does not imply a decrease in overall production capabilities; rather, it defines a point on a production curve at which producing an additional unit of output will result in a lower profit. Under diminishing returns, output remains positive, but productivity and efficiency decrease. The modern understanding of the law adds the dimension of holding other outputs equal, since a given pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular Set Function
In mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit ( diminishing returns). The natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set of \Omega, which satisfies one of the following equivalent condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
