Continuous Convex Research Articles

Simple phase unwrapping method with continuous convex minimization : erratum (Optics Express, (2022), Vol. 30, Issue 18, p33395-33411)

Simple phase unwrapping method with continuous convex minimization : erratum (Optics Express, (2022), Vol. 30, Issue 18, p33395-33411)

Resource optimization of multi-UAV assisted communication system based on user scheduling

In order to improve the transmission rate of the multi-user mobile communication downlink wireless transmission system, a resource optimization method for multi-unmanned aerial vehicle (UAV) assisted communication system based on user scheduling and trajectory optimization is proposed. The proposed method establishes an optimization problem with the maximization of the total multi-user throughput as the criterion under the constraints of user scheduling, total energy consumption of UAVs and user service quality requirements. In order to solve this non-convex problem, the original non-convex problem is decomposed into three easy-to-handle non-convex sub-problems by using the block coordinate descent method (BCD). The sub-problems are transformed and solved by introducing slack variables, first-order Taylor expressions, continuous convex approximation (SCA) and other methods, and then alternately iteratively optimized to obtain an approximate suboptimal solution to the original non-convex problem. Simulation results show that the proposed algorithm can effectively improve the total system throughput and has good convergence in both single and multi-UAV communication systems.

Concave/Convex Curvature of Anisotropic Grooves Differentially Alters Cellular Morphology, Adhesion, and Proliferation.

Curvature is an integral part of the complex in vivo tissue architecture across various length scales. Therefore, several in vitro models with a patterned curvature in different length scales have been developed to understand the role of this in cellular behavior. At the subcellular scale, wavy patterns have been reported wherein concave and convex grooves are adjacently present. However, the independent effect of continuous subcellular concave and convex shapes has not been reported, mainly owing to the limitations in fabricating such patterns. In this study, we developed continuous concave and convex grooves on polydimethylsiloxane (PDMS) using a Dracaena sanderiana (bamboo) leaf as a template. The first (negative) replica from the abaxial side of the bamboo leaf, which imparted concave grooves on PDMS, was subsequently used as a template to fabricate a positive replica of the leaf, resulting in convex grooves of the same size and arrangement as the concave grooves. We examined the influence of the groove curvature on the morphology of bone marrow-derived human mesenchymal stem cells (BM-hMSCs) and skeletal muscle cells (C2C12). BM-hMSCs and C2C12 cells aligned on both concave and convex grooves as compared to the random orientation on a flat substrate. The significant difference was observed in the morphology of both cells, in terms of area, aspect ratio, number, and length of protrusions on concave and convex patterns. We found that the number of protrusions was also dependent on the ratio of cell to pattern length scale for convex-shaped grooves but independent of length scale for concave-shaped grooves. The proliferation of BM-hMSCs was also found to be different on concave and convex shapes. Therefore, this study shows the importance of (1) convex and concave curvatures of the subcellular length scale in cellular response, (2) dependence on the ratio of cell and curvature length scale, and (3) use of natural templates for overcoming fabrication challenges.

On locally Lipschitz convex functions defined on subsets with empty interior of locally convex spaces

The generic Fréchet and/or Gateaux differentiability of a continuous convex function on an open convex subset of a Banach space is well studied and has important theoretical implications. An example of Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778] shows that similar results are not true when the openness of the domain is not ensured, but such properties can be obtained if the respective convex function is assumed to be locally Lipschitz, as shown by Verona [More on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(1):137–140]; Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778]; Noll [Generic Gâteaux-differentiability of convex functions on small sets. J Math Anal Appl. 1990;147(2):531–544] and others. It is our aim to extend such results for convex functions defined on (not necessarily open) convex subsets of locally convex spaces. In the same framework we extend Gale's duality theorem (1967), as well as a result of H. Ergin & T. Sarver on the unique dual representation of a convex function (2010). Moreover, a detailed study of the Rainwater example is done.

Robust secure resource optimization for active STAR-RIS systems

In this paper, we investigate the robust secure resource optimization for the active simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) system under the imperfect eavesdroppers channel state information. Considering the fairness, a max–min secure rate optimization problem is formulated based on several practical constraints. To deal with the original non-convex problem, an alternative iteration algorithm is proposed. First, the original problem is decomposed into two non-convex sub-problems. Next, the continuous convex approximation and S-procedure techniques are applied to deal with the non-convex and uncertainty constraints, respectively. Then, the first-order Taylor expansion formula is utilized to approximate the convex difference form of the objective function. Finally, two sub-problems are transformed into convex ones and alternately solved until convergence. Simulation results show that the secure rate of the proposed scheme is higher than the conventional schemes.

Non-orthogonal multiple access-based MEC for energy-efficient task offloading in e-commerce systems

Mobile edge computing (MEC) reduces the latency for end users to access applications deployed at the edge by offloading tasks to the edge. With the popularity of e-commerce and the expansion of business scale, server load continues to increase, and energy efficiency issues gradually become more prominent. Computation offloading has received widespread attention as a technology that effectively reduces server load. However, how to improve energy efficiency while ensuring computing requirements is an important challenge facing computation offloading. To solve this problem, using non-orthogonal multiple access (NOMA) to increase the efficiency of multi-access wireless transmission, MEC supporting NOMA is investigated in the research. Computing resources will be divided into separate sub-computing that will be handled via e-commerce terminals or transferred to edge sides by reutilizing radio resources, we put forward a Group Switching Matching Algorithm Based on Resource Unit Allocation (GSM-RUA) algorithm that is multi-dimensional. To this end, we first formulate this task allocation problem as a long-term stochastic optimization problem, which we then convert to three short-term deterministic sub-programming problems using Lyapunov optimization, namely, radio resource allocation in a large timescale, computation resource allocating and splitting in a small-time frame. Of the 3 short-term deterministic sub-programming problems, the first sub-programming problem can be remodeled into a 1 to n matching problem, which can be solved using the block-shift-matching-based radio resource allocation method. The latter two sub-programming problems are then transformed into two continuous convex problems by relaxation and then solved easily. We then use simulations to prove that our GSM-RUA algorithm is superior to the state-of-the-art resource management algorithms in terms of energy consumption, efficiency and complexity for e-commerce scenarios.

New Auxiliary Principle Technique for General Harmonic Directional Variational Inequalities

This paper explores the utilisation of harmonic variational inequalities to establish the minimum value among two locally Lipschitz continuous harmonic convex functions. This investigation introduces novel classes of harmonic directed variational inequalities, particularly focusing on scenarios like harmonic complementarity and related optimization challenges. The study proposes and analyses various inertial iterative strategies for addressing harmonic directed variational inequalities through the auxiliary principle technique. It examines convergence criteria under specific weak conditions, emphasising the simplicity of the approach compared to other methodologies. The findings presented herein have broad applicability in the context of harmonic variational inequalities and optimization problems, though they are limited to theoretical exploration. Further research is required to implement these strategies numerically.

On the properties of one auxiliary function for calculating the global extremum

In this paper, we investigate the properties of one Auxiliary Function (AF) for calculating the global optimum of a function of many variables. The considered AF is constructed by transforming the objective function using the Lebesgue integral and is a function of one variable. Despite the fact that in previous works this AF was used to find the global minimum of smooth functions on convex closed sets using the algorithm of dividing a segment in half, its properties have not been studied. Here, this AF is used to calculate the global extremum of continuous functions defined on bounded closed sets of a multidimensional Euclidean space. The basic properties of the AF for any continuous objective function are established, such as non-negativity, positive uniformity, uniform continuity, differentiability and strict convexity, moreover, derivatives up to a certain order are found. The optimality criterion is formulated. The main criterion of optimality is that the value of a free variable, at which the AF and its derivatives are equal to zero up to some order, coincides with the global minimum of the objective function. It follows from this optimality criterion that to calculate the global minimum of the objective function, it is sufficient to find the zero of the AF or its derivative up to some order.

Smoothing algorithms for nonsmooth optimization over the Stiefel manifold with applications to the graph Fourier basis problem

In this paper, we consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with a linear mapping. Besides, we are interested in its application to the graph Fourier basis problem. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms.

First-Order Algorithms Without Lipschitz Gradient: A Sequential Local Optimization Approach

Most first-order methods rely on the global Lipschitz continuity of the objective gradient, which fails to hold in many problems. This paper develops a sequential local optimization (SLO) framework for first-order algorithms to optimize problems without Lipschitz gradient. Operating on the assumption that the gradient is locally Lipschitz continuous over any compact set, SLO develops a careful scheme to control the distance between successive iterates. The proposed framework can easily adapt to the existing first-order methods, such as projected gradient descent (PGD), truncated gradient descent (TGD), and a parameter-free variant of Armijo linesearch. We show that SLO requires [Formula: see text] gradient evaluations to find an ϵ-stationary point, where Y is certain compact set with [Formula: see text] radius, and [Formula: see text] denotes the Lipschitz constant of the i-th order derivatives in Y. It is worth noting that our analysis provides the first nonasymptotic convergence rate for the (slight variant of) Armijo linesearch algorithm without globally Lipschitz continuous gradient or convexity. As a generic framework, we also show that SLO can incorporate more complicated subroutines, such as a variant of the accelerated gradient descent (AGD) method that can harness the problem’s second-order smoothness without Hessian computation, which achieves an improved [Formula: see text] complexity. Funding: J. Zhang is supported by the MOE AcRF [Grant A-0009530-04-00], from Singapore Ministry of Education. M. Hong is supported by NSF [Grants CIF-1910385 and EPCN-2311007]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2021.0029 .

Jointly Active/Passive Beamforming Optimization for Intelligent-Reflecting Surface-Assisted Cognitive-IoT Networks

To overcome challenges such as limited energy availability for terminal devices, constrained network coverage, and suboptimal spectrum resource utilization, with the overarching objective of establishing a sustainable and efficient interconnection infrastructure, we introduce an innovative Intelligent Reflective Surface (IRS) technology. This cutting-edge IRS technology is employed to architect a wireless and energy-efficient cognitive secure communication network assisted by IRS. To further optimize the overall energy harvesting of this network, we present a cognitive secure resource allocation scheme, aiming to maximize the system’s total collected energy. This scheme carefully considers various constraints, including transmission power constraints for cognitive base stations, power constraints for jammer devices, interference limitations for all primary users, minimum security rate constraints for all cognitive Internet of Things (IoT) devices, and phase shift constraints for IRS. We establish a comprehensive hybrid cognitive secure resource allocation model, encompassing joint cognitive transmission beam design, jammer device transmission beam design, and phase shift design. Given the non-convex nature of the formulated problem and the intricate coupling relationships among variables, we devise an effective block coordinate descent (BCD) iterative algorithm. The realization of joint cognitive/jammer base station transmission beam design and phase shift design employs sophisticated techniques such as continuous convex approximation methods and semi-definite programming. Simulation results underscore the superior performance of the proposed scheme compared to existing resource allocation approaches, particularly in terms of total harvested energy and other critical metrics.

Optimizing V2X Communication: Spectrum Resource Allocation and Power Control Strategies for Next-Generation Wireless Technologies

The upcoming wireless technology developments in the next generations are expected to substantially transform the vehicle-to-everything (V2X) communication network. The challenge of limited spectrum resources in V2X communication, caused by the need for high data rates, necessitates a thorough analysis of spectrum resource allocation and power control. This complex problem falls under the domain of mixed-integer nonlinear programming; a strategic approach is implemented to overcome these issues, which divides the main challenge into two sub-problems. The issue of resource allocation is addressed by implementing a multiaccess spectrum allocation method, which is deliberately designed to optimize the utilization of the spectrum resources that are currently accessible. Concurrently, the power control issue is resolved by employing a continuous convex approximation technique, which effectively converts non-convex power-allocation issues into convex equivalents. This approach helps to alleviate interference between users. Finally, the simulation results prove that the proposed approaches can improve vehicle performance. The algorithms proposed in this article significantly improve the system throughput and access rate of vehicular user equipment (VUEs) while ensuring the data rate of cellular user equipment (CUEs).

Alternated inertial forward-backward-forward splitting algorithm

In this paper, we propose a forward-backward-forward splitting algorithm with alternated inertial step for finding a zero of the sum of two monotone operators in a real Hilbert space. Then, by making use of primal-dual techniques we derive the alternated inertial primal-dual splitting algorithm of forward-backward-forward type for solving structured monotone inclusion problems involving parallel sums and compositions of maximally monotone operators with linear continuous ones and convex minimization problems. Finally, we demonstrate numerically that our proposed algorithm performs better than the existing algorithms.

A new continuous variables and convex relaxation based formulation for reconfiguration of unbalanced mutually coupled active distribution networks

With the integration of renewable energy-based sources and demand response initiatives, the distribution network has transformed into an active network with bidirectional flows. Network Reconfiguration (NR), i.e., changing network topology, is an alternative to minimize power loss in the distribution network with sectionalizing switches (normally closed) and tie-line switches (normally open). Due to the discrete nature of switches (on or off), the NR formulation is generally a mixed-integer NP-hard nonlinear problem. This paper proposes the first NR formulation for an unbalanced active distribution network with continuous variables to indicate switches’ statuses in NR while considering mutual coupling between line impedances. The obtained solution provides the optimal real power and voltage magnitude settings of dispatchable Distributed Generators (DGs) and network topology for the minimum active power loss in the network. The proposed formulation considers all practical aspects of the distribution network, i.e., mutual coupling, load unbalances, angular unbalance, and the absence of one or two phases in a line section, which, otherwise, have not been considered together in previous formulations. Further, a novel convex relaxation for bus voltage phase angles avoids explicit voltage recovery. Compared to prior formulations, numeric results for unbalanced radial distribution networks showcase the proposed formulation’s effectiveness.

A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

AbstractThis paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures.

A computationally efficient sequential convex programming using Chebyshev collocation method

This paper aims to develop a computationally efficient sequential convex programming algorithm for a class of nonlinear systems. A Chebyshev collocation discretization (CCD) technique is proposed that transforms the continuous convex optimization problem into a finite-dimensional discrete optimization problem. The CCD technique uses Chebyshev polynomials to construct a closed-form approximate solution to the convex dynamic equation. Moreover, it establishes a linear mapping between the control inputs and the system states. This enables the constraints of the dynamics equations to be externalized from the optimization problem and computed using efficient linear equation solving algorithms. On the one hand, this strategy significantly reduces the number of constraint equations and optimization variables, improving the speed of the optimizer. On the other hand, it preserves the path constraints on the system states in the optimization problem. A numerical simulation example of the perching maneuver for a fixed-wing unmanned aerial vehicle is presented to validate the efficiency of the algorithm. The result shows significant improvements in both speed and accuracy compared to the Euler discretization, and greatly improves the solution speed while maintaining almost the same solution accuracy compared to the Gauss pseudospectral optimization method.

A hybrid patch decomposition approach to compute an enclosure for multi-objective mixed-integer convex optimization problems

In multi-objective mixed-integer convex optimization, multiple convex objective functions need to be optimized simultaneously while some of the variables are restricted to take integer values. In this paper, we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization problem into several multi-objective continuous convex optimization problems, which we refer to as patches. We then dynamically compute and improve coverages of the nondominated sets of those patches to finally combine them to obtain an enclosure of the nondominated set of the multi-objective mixed-integer convex optimization problem. Additionally, we introduce a mechanism to reduce the number of patches that need to be considered in total. Our new algorithm is the first of its kind and guaranteed to return an enclosure of prescribed quality within a finite number of iterations. For selected numerical test instances we compare our new criterion space based approach to other algorithms from the literature and show that much larger instances can be solved with our new algorithm.

ON SEPARATE CONTINUITY AND SEPARATE CONVEXITY: A SYNTHETIC TREATMENT FOR FUNCTIONS AND SETS

AbstractThis paper relies on nested postulates of separate, linear and arc-continuity of functions to define analogous properties for sets that are weaker than the requirement that the set be open or closed. This allows three novel characterisations of open or closed sets under convexity or separate convexity postulates: the first pertains to separately convex sets, the second to convex sets and the third to arbitrary subsets of a finite-dimensional Euclidean space. By relying on these constructions, we also obtain new results on the relationship between separate and joint continuity of separately quasiconcave, or separately quasiconvex functions. We present examples to show that the sufficient conditions we offer cannot be dispensed with.

Extendability of continuous quasiconvex functions from subspaces

Let Y be a subspace of a topological vector space X, and A⊂X an open convex set that intersects Y. We say that the property (QE) [property (CE)] holds if every continuous quasiconvex [continuous convex] function on A∩Y admits a continuous quasiconvex [continuous convex] extension defined on A. We study relations between (QE) and (CE) properties, proving that (QE) always implies (CE) and that, under suitable hypotheses (satisfied for example if X is a normed space and Y is a closed subspace of X), the two properties are equivalent. By combining the previous implications between (QE) and (CE) properties with known results about the property (CE), we obtain some new positive results about the extension of quasiconvex continuous functions. In particular, we generalize the results contained in [9] to the infinite-dimensional separable case. Moreover, we also immediately obtain existence of examples in which (QE) does not hold.

A New Distributed Robust Power Control for Two-Layer Cooperative Communication Networks in Smart Grids with Reduced Utility Costs

The packet loss during transmission of load control commands can lead to regulation errors in the smart grid and increase the cost of utility agencies due to the purchase of additional automatic generation control (AGC) services. In this paper, a two-layer cooperative communication network between the utility company and relays is presented. The utility company rents the relay to assist with the downlink transmission to improve the reliability of communication and reduce the data transmission cost due to packet loss. Furthermore, the uncertainty of channel gain is considered, and a two-tier game model is established. A distributed robust power control algorithm based on the continuous convex approximation method is proposed to obtain the optimal relay power allocation and price. Through the simulation analysis of the proposed scheme and the two comparison schemes, the cost of the utility company was reduced by 6% and 21%, and the standard deviation of income value between the relays was reduced by 40% and 48%, respectively.