In classical nonlinear programming, it is a general method of developing optimality conditions that a nonlinear programming problem is linearized as a linear programming problem by using first order approximations of ...In classical nonlinear programming, it is a general method of developing optimality conditions that a nonlinear programming problem is linearized as a linear programming problem by using first order approximations of the functions at a given feasible point. The linearized procedure for differentiable nonlinear programming problems can be naturally generalized to the quasi differential case. As in classical case so called constraint qualifications have to be imposed on the constraint functions to guarantee that for a given local minimizer of the original problem the nullvector is an optimal solution of the corresponding 'quasilinearized' problem. In this paper, constraint qualifications for inequality constrained quasi differentiable programming problems of type min {f(x)|g(x)≤0} are considered, where f and g are qusidifferentiable functions in the sense of Demyanov. Various constraint qualifications for this problem are presented and a new one is proposed. The relations among these conditions are investigated. Moreover, a Wolf dual problem for this problem is introduced, and the corresponding dual theorems are given.展开更多
In this paper, we consider the composed convex optimization problem which consists in minimizing the sum of a convex function and a convex composite function. By using the properties of the epigraph of the conjugate f...In this paper, we consider the composed convex optimization problem which consists in minimizing the sum of a convex function and a convex composite function. By using the properties of the epigraph of the conjugate functions and the subdifferentials of convex functions, we give some new constraint qualifications which completely characterize the strong Fenchel duality and the total Fenchel duality for composed convex optimiztion problem in real locally convex Hausdorff topological vector spaces.展开更多
In this paper, we derive an exact penalty function for nonconvex bilevel programming problem based on its KS form. Based on this exact penalty function a sufficient condition for KS to be partially calm is presented a...In this paper, we derive an exact penalty function for nonconvex bilevel programming problem based on its KS form. Based on this exact penalty function a sufficient condition for KS to be partially calm is presented and a necessary optimality condition for nonconvex bilevel programming problems is given. Some existing results about the differentiability of the value function of the lower level programming problem are extended and a sufficient condition for CRCQ to hold for VS form of BLPP with linear lower level programming problem is also given.展开更多
Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions...Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.展开更多
Numeracy is the capacity to use mathematical ideas in all facets of life.It involves activities such as adding and subtracting numbers,counting,number recognition,solving number problems involving various operations,s...Numeracy is the capacity to use mathematical ideas in all facets of life.It involves activities such as adding and subtracting numbers,counting,number recognition,solving number problems involving various operations,sorting,observing,identifying,and establishing patterns.It is one of the fundamental skills that students should have mastered by the end of their primary schooling.With the notable importance of mastery of numeracy skills,low achievement and performance of the learners were observed in this aspect.This study aimed in enhancing the numeracy skills of Grade 3 learners through authentic performance tasks.The variable in numeracy skills includes the four fundamental operations and problem solving.The quasi-experimental design was utilized wherein purposive sampling or non-randomized sampling was used.In this study,33 Grade 3 learners of Rizal Elementary School were selected to participate in the tests.Pre-test and post-test crafted by the teacher were the main instrument in the study.The result revealed that in the pre-test the learners obtained a mean percentage score(MPS)of 38.20%in four fundamental operations,which implied a non-numerate level.While in terms of problem solving,the learners obtained a MPS of 20.60%which is also in the non-numerate level.It has a grand mean of 29.40%with an interpretation of non-numerate level.In the post-test,it was observed that four fundamental operations have a MPS of 81.10%which is in average numerate level,while problem solving has a MPS of 76.30%with a grand mean of 78.70%with an interpretation of average numerate level.This implied that there is a significant difference between the pre-test and post-test in the four fundamental operations and problem solving.Thus,it can be concluded that the application of authentic performance tasks was effective to bridge the gap on numeracy skills.展开更多
文摘In classical nonlinear programming, it is a general method of developing optimality conditions that a nonlinear programming problem is linearized as a linear programming problem by using first order approximations of the functions at a given feasible point. The linearized procedure for differentiable nonlinear programming problems can be naturally generalized to the quasi differential case. As in classical case so called constraint qualifications have to be imposed on the constraint functions to guarantee that for a given local minimizer of the original problem the nullvector is an optimal solution of the corresponding 'quasilinearized' problem. In this paper, constraint qualifications for inequality constrained quasi differentiable programming problems of type min {f(x)|g(x)≤0} are considered, where f and g are qusidifferentiable functions in the sense of Demyanov. Various constraint qualifications for this problem are presented and a new one is proposed. The relations among these conditions are investigated. Moreover, a Wolf dual problem for this problem is introduced, and the corresponding dual theorems are given.
基金Supported by the National Natural Science Foundation of China(No.11461027)Hunan Provincial Natural Science Foundation of China(No.2016JJ2099)the Scientific Research Fund of Hunan Provincial Education Department(No.17A172)
文摘In this paper, we consider the composed convex optimization problem which consists in minimizing the sum of a convex function and a convex composite function. By using the properties of the epigraph of the conjugate functions and the subdifferentials of convex functions, we give some new constraint qualifications which completely characterize the strong Fenchel duality and the total Fenchel duality for composed convex optimiztion problem in real locally convex Hausdorff topological vector spaces.
文摘In this paper, we derive an exact penalty function for nonconvex bilevel programming problem based on its KS form. Based on this exact penalty function a sufficient condition for KS to be partially calm is presented and a necessary optimality condition for nonconvex bilevel programming problems is given. Some existing results about the differentiability of the value function of the lower level programming problem are extended and a sufficient condition for CRCQ to hold for VS form of BLPP with linear lower level programming problem is also given.
文摘Some remarks are made on the use of the Abadie constraint qualification, the Guignard constraint qualifications and the Guignard regularity condition in obtaining weak and strong Kuhn-Tucker type optimality conditions in differentiable vector optimization problems.
文摘Numeracy is the capacity to use mathematical ideas in all facets of life.It involves activities such as adding and subtracting numbers,counting,number recognition,solving number problems involving various operations,sorting,observing,identifying,and establishing patterns.It is one of the fundamental skills that students should have mastered by the end of their primary schooling.With the notable importance of mastery of numeracy skills,low achievement and performance of the learners were observed in this aspect.This study aimed in enhancing the numeracy skills of Grade 3 learners through authentic performance tasks.The variable in numeracy skills includes the four fundamental operations and problem solving.The quasi-experimental design was utilized wherein purposive sampling or non-randomized sampling was used.In this study,33 Grade 3 learners of Rizal Elementary School were selected to participate in the tests.Pre-test and post-test crafted by the teacher were the main instrument in the study.The result revealed that in the pre-test the learners obtained a mean percentage score(MPS)of 38.20%in four fundamental operations,which implied a non-numerate level.While in terms of problem solving,the learners obtained a MPS of 20.60%which is also in the non-numerate level.It has a grand mean of 29.40%with an interpretation of non-numerate level.In the post-test,it was observed that four fundamental operations have a MPS of 81.10%which is in average numerate level,while problem solving has a MPS of 76.30%with a grand mean of 78.70%with an interpretation of average numerate level.This implied that there is a significant difference between the pre-test and post-test in the four fundamental operations and problem solving.Thus,it can be concluded that the application of authentic performance tasks was effective to bridge the gap on numeracy skills.
