Transversality, regularity and error bounds in variational analysis and optimisation
- Authors: Nguyen, Duy
- Date: 2021
- Type: Text , Thesis , PhD
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- Description: Transversality properties of collections of sets, regularity properties of set-valued mappings, and error bounds of extended-real-valued functions lie at the core of variational analysis because of their importance for stability analysis, constraint qualifications, qualification conditions in coderivative and subdifferential calculus, and convergence analysis of numerical algorithms. The thesis is devoted to investigation of several research questions related to the aforementioned properties. We develop a general framework for quantitative analysis of nonlinear transversality properties by establishing primal and dual characterizations of the properties in both convex and nonconvex settings. The H¨older case is given special attention. Quantitative relations between transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space are also discussed. We study a new property so called semitransversality of collections of set-valued mappings on metric (in particular, normed) spaces. The property is a generalization of the semitransversality of collections of sets and the negation of the corresponding stationarity, a weaker property than the extremality of collections of set-valued mappings. Primal and dual characterizations of the property as well as quantitative relations between the property and semiregularity of set-valued mappings are formulated. As a consequence, we establish dual necessary and sufficient conditions for stationarity of collections of set-valued mappings as well as optimality conditions for efficient solutions with respect to variable ordering structures in multiobjective optimization. We examine a comprehensive (i.e. not assuming the mapping to have any particular structure) view on the regularity theory of set-valued mappings and clarify the relationships between the existing primal and dual quantitative sufficient and necessary conditions including their hierarchy. The typical sequence of regularity assertions, often hidden in the proofs, and the roles of the assumptions involved in the assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund are exposed. As a consequence, we formulate primal and dual conditions for the stability properties of solution mappings to inclusions. We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. As a consequence, the error bound theory is applied to characterize subregularity of set-valued mappings, and calmness of the solution mapping in convex semi-infinite optimization problems.
- Description: Doctor of Philosophy
Energy-efficient design for downlink cloud radio access networks
- Authors: Vu, Tung , Ngo, Duy , Dao, Minh , Durrani, Salman , Nguyen, Duy , Middleton, Richard
- Date: 2018
- Type: Text , Conference proceedings
- Relation: 2018 IEEE International Conference on Communications (ICC); Kansas City, MO, USA; 20-24 May 2018 p. 1-6
- Full Text: false
- Reviewed:
- Description: This work aims to maximize the energy efficiency of a downlink cloud radio access network (C-RAN), where data is transferred from a baseband unit in the core network to several remote radio heads via a set of edge routers over capacity-limited fronthaul links. The remote radio heads then send the received signals to their users via radio access links. We formulate a new mixed-integer nonlinear problem in which the ratio of network throughput and total power consumption is maximized. This challenging problem formulation includes practical constraints on routing, predefined minimum data rates, fronthaul capacity and maximum RRH transmit power. By employing the successive convex quadratic programming framework, an iterative algorithm is proposed with guaranteed convergence to a Fritz John solution of the formulated problem. Significantly, each iteration of the proposed algorithm solves only one simple convex program. Numerical examples with practical parameters confirm that the proposed joint optimization design markedly improves the C-RAN's energy efficiency compared to benchmark schemes.
Energy efficiency maximization for downlink cloud radio access networks with data sharing and data compression
- Authors: Tung Thanh, Vu , Duy Trong, Ngo , Dao, Minh , Durrani, Salman , Nguyen, Duy , Middleton, Richard
- Date: 2018
- Type: Text , Journal article
- Relation: IEEE transactions on wireless communications Vol. 17, no. 8 (2018), p. 4955-4970
- Full Text: false
- Reviewed:
- Description: This paper aims to maximize the energy efficiency of a downlink cloud radio access network (C-RAN). Here, data is transferred from a baseband unit in the core network to several remote radio heads via a set of edge routers over capacity-limited fronthaul links. The remote radio heads then send the received signals to their users via radio access links. Both data sharing and compression-based strategies are considered for fronthaul data transfer. New mixed-integer nonlinear problems are formulated, in which the ratio of network throughput and total power consumption is maximized. These challenging problem formulations include practical constraints on routing, predefined minimum data rates, fronthaul capacity, and maximum remote radio head transmit power. By employing the successive convex quadratic programming, iterative algorithms are proposed with guaranteed convergence to the Fritz John solutions of the formulated problems. Significantly, each iteration of the proposed algorithms solves only one simple convex program. Numerical examples with practical parameters confirm that the proposed joint optimization designs markedly improve the C-RAN's energy efficiency compared to benchmark schemes. They also show that the fronthaul data-sharing strategy outperforms its compression-based counterpart in terms of energy efficiency, in both single-hop and multi-hop network scenarios.