The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.
{"title":"Optimality conditions and constraint qualifications for cardinality constrained optimization problems","authors":"Zhuo-Feng Xiao, J. Ye","doi":"10.3934/naco.2023011","DOIUrl":"https://doi.org/10.3934/naco.2023011","url":null,"abstract":"The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"30 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83407789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the deho- mogenization approach.
{"title":"Dehomogenization for completely positive tensors","authors":"Jiawang Nie, Xindong Tang, Zi Yang, Suhan Zhong","doi":"10.3934/naco.2022037","DOIUrl":"https://doi.org/10.3934/naco.2022037","url":null,"abstract":". A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the deho- mogenization approach.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79664111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper establishes that a class of $N$-player stochastic games with singular controls, either of bounded velocity or of finite variation, can both be approximated by mean field games (MFGs) with singular controls of bounded velocity. More specifically, it shows (i) the optimal control to an MFG with singular controls of a bounded velocity $theta$ is shown to be an $epsilon_N$-NE to an $N$-player game with singular controls of the bounded velocity, with $epsilon_N = O(frac{1}{sqrt{N}})$, and (ii) the optimal control to this MFG is an $(epsilon_N + epsilon_{theta})$-NE to an $N$-player game with singular controls of finite variation, where $epsilon_{theta}$ is an error term that depends on $theta$. This work generalizes the classical result on approximation $N$-player games by MFGs, by allowing for discontinuous controls.
{"title":"Approximation of $ N $-player stochastic games with singular controls by mean field games","authors":"Haoyang Cao, Xin Guo, Joon Seok Lee","doi":"10.3934/naco.2023001","DOIUrl":"https://doi.org/10.3934/naco.2023001","url":null,"abstract":"This paper establishes that a class of $N$-player stochastic games with singular controls, either of bounded velocity or of finite variation, can both be approximated by mean field games (MFGs) with singular controls of bounded velocity. More specifically, it shows (i) the optimal control to an MFG with singular controls of a bounded velocity $theta$ is shown to be an $epsilon_N$-NE to an $N$-player game with singular controls of the bounded velocity, with $epsilon_N = O(frac{1}{sqrt{N}})$, and (ii) the optimal control to this MFG is an $(epsilon_N + epsilon_{theta})$-NE to an $N$-player game with singular controls of finite variation, where $epsilon_{theta}$ is an error term that depends on $theta$. This work generalizes the classical result on approximation $N$-player games by MFGs, by allowing for discontinuous controls.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77342384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decompositions and Choquet convex decompositions, as well as their corresponding hull operators acting on the power sets of Lebesgue-Bochner spaces. We show that Choquet hull coincides with convex hull in the finite-dimensional setting, yet Choquet hull tends to be larger in infinite dimensions. We also provide a quantitative characterization of Choquet hull. Furthermore, we show that Choquet decomposable hull of a set coincides with its (strongly) closed decomposable hull and the Choquet convex decomposable hull of a set coincides with its Choquet decomposable hull of the convex hull. It turns out that the collection of all measurable selections of a closed-valued multifunction is Choquet decomposable and those of a closed convex-valued multifunction is Choquet convex decomposable. Finally, we investigate the operator-type features of Choquet decomposable and Choquet convex decomposable hull operators when applied in succession.
{"title":"Random sets and Choquet-type representations","authors":"cCaugin Ararat, Umur Cetin","doi":"10.3934/naco.2023008","DOIUrl":"https://doi.org/10.3934/naco.2023008","url":null,"abstract":"As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decompositions and Choquet convex decompositions, as well as their corresponding hull operators acting on the power sets of Lebesgue-Bochner spaces. We show that Choquet hull coincides with convex hull in the finite-dimensional setting, yet Choquet hull tends to be larger in infinite dimensions. We also provide a quantitative characterization of Choquet hull. Furthermore, we show that Choquet decomposable hull of a set coincides with its (strongly) closed decomposable hull and the Choquet convex decomposable hull of a set coincides with its Choquet decomposable hull of the convex hull. It turns out that the collection of all measurable selections of a closed-valued multifunction is Choquet decomposable and those of a closed convex-valued multifunction is Choquet convex decomposable. Finally, we investigate the operator-type features of Choquet decomposable and Choquet convex decomposable hull operators when applied in succession.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91384043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jin Ma has made fundamental contributions to the theory of forward-backward stochastic differential equations (FBSDEs, for short). In this paper, as one of his main collaborators, I will recall the initiation, the path of the development for FBSDEs, as well as some interesting historic stories.
{"title":"Forward-backward stochastic differential equations: Initiation, development and beyond","authors":"J. Yong","doi":"10.3934/naco.2022011","DOIUrl":"https://doi.org/10.3934/naco.2022011","url":null,"abstract":"<p style='text-indent:20px;'>Jin Ma has made fundamental contributions to the theory of forward-backward stochastic differential equations (FBSDEs, for short). In this paper, as one of his main collaborators, I will recall the initiation, the path of the development for FBSDEs, as well as some interesting historic stories.</p>","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"40 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88483597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a new projection method (extragradient method) for solving the generalized variational inequality problem. Our method is well-defined and is proven to be globally convergent to a solution of the problem under the assumptions that the multivalued mapping is continuous and satisfies a condition that is strictly weaker than the pseudomonotonicity. Finally, to support our results and to illustrate the numerical behavior of the proposed algorithm, some fundamental experiments are provided and also compared with recent works.
{"title":"An improved projection algorithm for variational inequality problem with multivalued mapping","authors":"Ouafa Belguidoum, Hassina Grar","doi":"10.3934/naco.2022002","DOIUrl":"https://doi.org/10.3934/naco.2022002","url":null,"abstract":"In this paper, we propose a new projection method (extragradient method) for solving the generalized variational inequality problem. Our method is well-defined and is proven to be globally convergent to a solution of the problem under the assumptions that the multivalued mapping is continuous and satisfies a condition that is strictly weaker than the pseudomonotonicity. Finally, to support our results and to illustrate the numerical behavior of the proposed algorithm, some fundamental experiments are provided and also compared with recent works.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"70 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76242225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study regression modeling is proposed for calculating the heating and cooling load of the buildings by considering the outdoor meteorological data. For this pupose, Balikesir University Rectorate Building is selected as the case building. In the winter months, measurements were made in the hot water boiler in the basement of the building for the heating energy load. For the cooling energy load, measurements were made in the chiller groups near the building. As climate data, eight variables were considered: outdoor temperature, solar radiation, relative humidity, wind speed, atmospheric pressure, sunshine duration, steam pressure, and 1 m underground temperature. The Minitab statistical analysis program was used to perform the modeling. 55 samples were used for mathematically modeling the heating load, while 37 samples are used for modeling the cooling load. The Rbegin{document}$ {}^{2 } $end{document}(coefficient of determination) values are calculated as 96.2% and 98.94% for cooling load and heating load, respectively. In addition to these findings, ANOVA results for both models were examined and both models were found to be significant.
In this study regression modeling is proposed for calculating the heating and cooling load of the buildings by considering the outdoor meteorological data. For this pupose, Balikesir University Rectorate Building is selected as the case building. In the winter months, measurements were made in the hot water boiler in the basement of the building for the heating energy load. For the cooling energy load, measurements were made in the chiller groups near the building. As climate data, eight variables were considered: outdoor temperature, solar radiation, relative humidity, wind speed, atmospheric pressure, sunshine duration, steam pressure, and 1 m underground temperature. The Minitab statistical analysis program was used to perform the modeling. 55 samples were used for mathematically modeling the heating load, while 37 samples are used for modeling the cooling load. The Rbegin{document}$ {}^{2 } $end{document}(coefficient of determination) values are calculated as 96.2% and 98.94% for cooling load and heating load, respectively. In addition to these findings, ANOVA results for both models were examined and both models were found to be significant.
{"title":"Modeling the relationship between outdoor meteorological data and energy consumptions at heating and cooling periods: Application in a university building","authors":"O. Kon, B. Yuksel, A. D. Karaoglan","doi":"10.3934/naco.2022021","DOIUrl":"https://doi.org/10.3934/naco.2022021","url":null,"abstract":"In this study regression modeling is proposed for calculating the heating and cooling load of the buildings by considering the outdoor meteorological data. For this pupose, Balikesir University Rectorate Building is selected as the case building. In the winter months, measurements were made in the hot water boiler in the basement of the building for the heating energy load. For the cooling energy load, measurements were made in the chiller groups near the building. As climate data, eight variables were considered: outdoor temperature, solar radiation, relative humidity, wind speed, atmospheric pressure, sunshine duration, steam pressure, and 1 m underground temperature. The Minitab statistical analysis program was used to perform the modeling. 55 samples were used for mathematically modeling the heating load, while 37 samples are used for modeling the cooling load. The Rbegin{document}$ {}^{2 } $end{document}(coefficient of determination) values are calculated as 96.2% and 98.94% for cooling load and heating load, respectively. In addition to these findings, ANOVA results for both models were examined and both models were found to be significant.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81476654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Canghua Jiang, Dongming Zhang, Chi Yuan, K. L. Teo
This paper proposes an active set solver for begin{document}$ H_infty $end{document} min-max optimal control problems involving linear discrete-time systems with linearly constrained states, controls and additive disturbances. The proposed solver combines Riccati recursion with dynamic programming. To deal with possible degeneracy (i.e. violations of the linear independence constraint qualification), constraint transformations are introduced that allow the surplus equality constraints on the state at each stage to be moved to the previous stage together with their Lagrange multipliers. In this way, degeneracy for a feasible active set can be determined by checking whether there exists an equality constraint on the initial state over the prediction horizon. For situations when the active set is degenerate and all active constraints indexed by it are non-redundant, a vertex exploration strategy is developed to seek a non-degenerate active set. If the sampled state resides in a robust control invariant set and certain second-order sufficient conditions are satisfied at each stage, then a bounded begin{document}$ l_2 $end{document} gain from the disturbance to controlled output can be guaranteed for the closed-loop system under some standard assumptions. Theoretical analysis and numerical simulations show that the computational complexity per iteration of the proposed solver depends linearly on the prediction horizon.
This paper proposes an active set solver for begin{document}$ H_infty $end{document} min-max optimal control problems involving linear discrete-time systems with linearly constrained states, controls and additive disturbances. The proposed solver combines Riccati recursion with dynamic programming. To deal with possible degeneracy (i.e. violations of the linear independence constraint qualification), constraint transformations are introduced that allow the surplus equality constraints on the state at each stage to be moved to the previous stage together with their Lagrange multipliers. In this way, degeneracy for a feasible active set can be determined by checking whether there exists an equality constraint on the initial state over the prediction horizon. For situations when the active set is degenerate and all active constraints indexed by it are non-redundant, a vertex exploration strategy is developed to seek a non-degenerate active set. If the sampled state resides in a robust control invariant set and certain second-order sufficient conditions are satisfied at each stage, then a bounded begin{document}$ l_2 $end{document} gain from the disturbance to controlled output can be guaranteed for the closed-loop system under some standard assumptions. Theoretical analysis and numerical simulations show that the computational complexity per iteration of the proposed solver depends linearly on the prediction horizon.
{"title":"An active set solver for constrained $ H_infty $ optimal control problems with state and input constraints","authors":"Canghua Jiang, Dongming Zhang, Chi Yuan, K. L. Teo","doi":"10.3934/naco.2021056","DOIUrl":"https://doi.org/10.3934/naco.2021056","url":null,"abstract":"This paper proposes an active set solver for begin{document}$ H_infty $end{document} min-max optimal control problems involving linear discrete-time systems with linearly constrained states, controls and additive disturbances. The proposed solver combines Riccati recursion with dynamic programming. To deal with possible degeneracy (i.e. violations of the linear independence constraint qualification), constraint transformations are introduced that allow the surplus equality constraints on the state at each stage to be moved to the previous stage together with their Lagrange multipliers. In this way, degeneracy for a feasible active set can be determined by checking whether there exists an equality constraint on the initial state over the prediction horizon. For situations when the active set is degenerate and all active constraints indexed by it are non-redundant, a vertex exploration strategy is developed to seek a non-degenerate active set. If the sampled state resides in a robust control invariant set and certain second-order sufficient conditions are satisfied at each stage, then a bounded begin{document}$ l_2 $end{document} gain from the disturbance to controlled output can be guaranteed for the closed-loop system under some standard assumptions. Theoretical analysis and numerical simulations show that the computational complexity per iteration of the proposed solver depends linearly on the prediction horizon.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82635824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we survey the primary research on the theory and applications of distributionally robust optimization (DRO). We start with reviewing the modeling power and computational attractiveness of DRO approaches, induced by the ambiguity sets structure and tractable robust counterpart reformulations. Next, we summarize the efficient solution methods, out-of-sample performance guarantee, and convergence analysis. Then, we illustrate some applications of DRO in machine learning and operations research, and finally, we discuss the future research directions.
{"title":"Distributionally Robust Optimization: A review on theory and applications","authors":"Fengming Lin, X. Fang, Zheming Gao","doi":"10.3934/naco.2021057","DOIUrl":"https://doi.org/10.3934/naco.2021057","url":null,"abstract":"In this paper, we survey the primary research on the theory and applications of distributionally robust optimization (DRO). We start with reviewing the modeling power and computational attractiveness of DRO approaches, induced by the ambiguity sets structure and tractable robust counterpart reformulations. Next, we summarize the efficient solution methods, out-of-sample performance guarantee, and convergence analysis. Then, we illustrate some applications of DRO in machine learning and operations research, and finally, we discuss the future research directions.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82948344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the exact boundary controllability for a degenerate and singular wave equation in a bounded interval with a moving endpoint. By the multiplier method and using an adapted Hardy-poincaré inequality, we prove direct and inverse inequalities for the solutions of the associated adjoint equation. As a consequence, by the Hilbert Uniqueness Method, we deduce the controllability result of the considered system when the control acts on the moving boundary. Furthermore, improved estimates of the speed of the moving endpoint and the controllability time are obtained.
{"title":"Exact controllability for a degenerate and singular wave equation with moving boundary","authors":"Alhabib Moumni, J. Salhi","doi":"10.3934/naco.2022001","DOIUrl":"https://doi.org/10.3934/naco.2022001","url":null,"abstract":"This paper is concerned with the exact boundary controllability for a degenerate and singular wave equation in a bounded interval with a moving endpoint. By the multiplier method and using an adapted Hardy-poincaré inequality, we prove direct and inverse inequalities for the solutions of the associated adjoint equation. As a consequence, by the Hilbert Uniqueness Method, we deduce the controllability result of the considered system when the control acts on the moving boundary. Furthermore, improved estimates of the speed of the moving endpoint and the controllability time are obtained.","PeriodicalId":44957,"journal":{"name":"Numerical Algebra Control and Optimization","volume":"28 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82617832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：