In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C 1.The paper deals with this problem by means of taking the place of maximum entropy function...In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C 1.The paper deals with this problem by means of taking the place of maximum entropy function with adjustable entropy function.By constructing an interval extension of adjustable entropy function an d some region deletion test rules,a new interval algorithm is presented.The rele vant properties are proven.The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum entropy algorithm.Both theoretical and numerica l results show that the method is reliable and efficient.展开更多
To solve the inequality problem, an adjustable entropy method is proposed. An inequality problem can be transformed into a minimax problem which is nondifferentiable; then an adjustable entropy is used to smooth the m...To solve the inequality problem, an adjustable entropy method is proposed. An inequality problem can be transformed into a minimax problem which is nondifferentiable; then an adjustable entropy is used to smooth the minimax problem. The solution of inequalities can be approached by using a BFGS algorithm of the standard optimization method. Some properties of the new approximate function are presented and then the global convergence are given according to the algorithm. Two numerical examples illustrate that the proposed method is efficient and is superior to the former ones.展开更多
This paper describes and explores a maximum-entropy approach to continuous minimax problem, which is applicable in many fields, such as transportation planning and game theory. It illustrates that the maximum entropy ...This paper describes and explores a maximum-entropy approach to continuous minimax problem, which is applicable in many fields, such as transportation planning and game theory. It illustrates that the maximum entropy approcach has easy framework and proves that every accumulation of {x_k} generated by maximum-entropy programming is -optimal solution of initial continuous minimax problem. The paper also explains BFGS or TR method for it. Two numerical exam.ples for continuous minimax problem are展开更多
Based on KKT complementary condition in optimization theory, an unconstrained non-differential optimization model for support vector machine is proposed. An adjustable entropy function method is given to deal with the...Based on KKT complementary condition in optimization theory, an unconstrained non-differential optimization model for support vector machine is proposed. An adjustable entropy function method is given to deal with the proposed optimization problem and the Newton algorithm is used to figure out the optimal solution. The proposed method can find an optimal solution with a relatively small parameter p, which avoids the numerical overflow in the traditional entropy function methods. It is a new approach to solve support vector machine. The theoretical analysis and experimental results illustrate the feasibility and efficiency of the proposed algorithm.展开更多
this paper,we introduce the L_(p) Shephard problem on entropy of log-concave functions,a comparison problem:whether ∏_(p)f≤∏_(p)g implies that Ent(f)≥Ent(g),for 1≤p<n,and Ent(f)≤Ent(g),for n<p,where ∏_(p)...this paper,we introduce the L_(p) Shephard problem on entropy of log-concave functions,a comparison problem:whether ∏_(p)f≤∏_(p)g implies that Ent(f)≥Ent(g),for 1≤p<n,and Ent(f)≤Ent(g),for n<p,where ∏_(p)f is the L_(p) projection body of a log-concave function f.Our results give a partial answer to this problem.展开更多
In non-smooth optimization,one particular problem which often appears inengineering designs,electrical engineering and game theory is called nonlinear minimaxproblem.For the non-smooth properties of objective function...In non-smooth optimization,one particular problem which often appears inengineering designs,electrical engineering and game theory is called nonlinear minimaxproblem.For the non-smooth properties of objective functions,there are some difficultiesin solving this problem.Since 1987,taking into account the entropy funtions,experts havehad several excellent results such as refs.[1—5].However,those methods are limited展开更多
The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is ...The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is affirmative if n≤4 and negative if n≥5.In this paper,we investigate the Busemann-Petty problem on entropy of log-concave functions:for even log-concave functions f and g with finite positive integrals in R^(n),if the marginal∫_(R^(n))∩H^(f(x)dx)of f is smaller than the marginal∫_(R^(n))∩H^(g(x)dx)of g for every hyperplane H passing through the origin,is the entropy Ent(f)of f bigger than the entropy Ent(g)of g?The BusemannPetty problem on entropy of log-concave functions includes the Busemann-Petty problem,and hence its answer is negative when n≥5.For 2≤n≤4,we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.展开更多
A nonlinear minimax problem is usually defined aswherefi(x), i=1,…,m, are generally smooth nonlinear functions of a vector x ∈ R^n. Since the objective φ(x) is a non-smooth function, (A) is then a non-smooth, uncon...A nonlinear minimax problem is usually defined aswherefi(x), i=1,…,m, are generally smooth nonlinear functions of a vector x ∈ R^n. Since the objective φ(x) is a non-smooth function, (A) is then a non-smooth, unconstrained optimization problem and cannot be solved by standard unconstrained minimization algorithms. One normally transforms it into an equivalent nonlinear programming problem:展开更多
Recently,the l_(p)minimization problem(p∈(0,1))for sparse signal recovery has been studied a lot because of its efficiency.In this paper,we propose a general smoothing algorithmic framework based on the entropy funct...Recently,the l_(p)minimization problem(p∈(0,1))for sparse signal recovery has been studied a lot because of its efficiency.In this paper,we propose a general smoothing algorithmic framework based on the entropy function for solving a class of l_(p)minimization problems,which includes the well-known unconstrained l_(2)-l_(p)problem as a special case.We show that any accumulation point of the sequence generated by the proposed algorithm is a stationary point of the l_(p)minimization problem,and derive a lower bound for the nonzero entries of the stationary point of the smoothing problem.We implement a specific version of the proposed algorithm which indicates that the entropy function-based algorithm is effective.展开更多
基金Supported by the National Natural Science Foundation of China(50 1 740 51 )
文摘In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C 1.The paper deals with this problem by means of taking the place of maximum entropy function with adjustable entropy function.By constructing an interval extension of adjustable entropy function an d some region deletion test rules,a new interval algorithm is presented.The rele vant properties are proven.The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum entropy algorithm.Both theoretical and numerica l results show that the method is reliable and efficient.
文摘To solve the inequality problem, an adjustable entropy method is proposed. An inequality problem can be transformed into a minimax problem which is nondifferentiable; then an adjustable entropy is used to smooth the minimax problem. The solution of inequalities can be approached by using a BFGS algorithm of the standard optimization method. Some properties of the new approximate function are presented and then the global convergence are given according to the algorithm. Two numerical examples illustrate that the proposed method is efficient and is superior to the former ones.
基金The Project was supported by National Natural Science Foundation of china.
文摘This paper describes and explores a maximum-entropy approach to continuous minimax problem, which is applicable in many fields, such as transportation planning and game theory. It illustrates that the maximum entropy approcach has easy framework and proves that every accumulation of {x_k} generated by maximum-entropy programming is -optimal solution of initial continuous minimax problem. The paper also explains BFGS or TR method for it. Two numerical exam.ples for continuous minimax problem are
基金the National Natural Science Foundation of China (60574075)
文摘Based on KKT complementary condition in optimization theory, an unconstrained non-differential optimization model for support vector machine is proposed. An adjustable entropy function method is given to deal with the proposed optimization problem and the Newton algorithm is used to figure out the optimal solution. The proposed method can find an optimal solution with a relatively small parameter p, which avoids the numerical overflow in the traditional entropy function methods. It is a new approach to solve support vector machine. The theoretical analysis and experimental results illustrate the feasibility and efficiency of the proposed algorithm.
基金The National Natural Science Foundation of China(11701373)The Shanghai Sailing Program(17YF1413800)。
文摘this paper,we introduce the L_(p) Shephard problem on entropy of log-concave functions,a comparison problem:whether ∏_(p)f≤∏_(p)g implies that Ent(f)≥Ent(g),for 1≤p<n,and Ent(f)≤Ent(g),for n<p,where ∏_(p)f is the L_(p) projection body of a log-concave function f.Our results give a partial answer to this problem.
基金Project supported by the National Natural Science Foundation of China.
文摘In non-smooth optimization,one particular problem which often appears inengineering designs,electrical engineering and game theory is called nonlinear minimaxproblem.For the non-smooth properties of objective functions,there are some difficultiesin solving this problem.Since 1987,taking into account the entropy funtions,experts havehad several excellent results such as refs.[1—5].However,those methods are limited
基金supported by National Natural Science Foundation of China(Grant No.12001291)supported by National Natural Science Foundation of China(Grant No.12071318)the Fundamental Research Funds for the Central Universities(Grant No.531118010593)。
文摘The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space R^(n) with smaller central hyperplane sections necessarily have smaller volumes.The solution has been completed and the answer is affirmative if n≤4 and negative if n≥5.In this paper,we investigate the Busemann-Petty problem on entropy of log-concave functions:for even log-concave functions f and g with finite positive integrals in R^(n),if the marginal∫_(R^(n))∩H^(f(x)dx)of f is smaller than the marginal∫_(R^(n))∩H^(g(x)dx)of g for every hyperplane H passing through the origin,is the entropy Ent(f)of f bigger than the entropy Ent(g)of g?The BusemannPetty problem on entropy of log-concave functions includes the Busemann-Petty problem,and hence its answer is negative when n≥5.For 2≤n≤4,we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.
基金Project supported by the National Natural Science Foundation of China
文摘A nonlinear minimax problem is usually defined aswherefi(x), i=1,…,m, are generally smooth nonlinear functions of a vector x ∈ R^n. Since the objective φ(x) is a non-smooth function, (A) is then a non-smooth, unconstrained optimization problem and cannot be solved by standard unconstrained minimization algorithms. One normally transforms it into an equivalent nonlinear programming problem:
基金supported by the National Natural Science Foundation of China(Nos.11171252,11431002).
文摘Recently,the l_(p)minimization problem(p∈(0,1))for sparse signal recovery has been studied a lot because of its efficiency.In this paper,we propose a general smoothing algorithmic framework based on the entropy function for solving a class of l_(p)minimization problems,which includes the well-known unconstrained l_(2)-l_(p)problem as a special case.We show that any accumulation point of the sequence generated by the proposed algorithm is a stationary point of the l_(p)minimization problem,and derive a lower bound for the nonzero entries of the stationary point of the smoothing problem.We implement a specific version of the proposed algorithm which indicates that the entropy function-based algorithm is effective.
