Optimal Approximation Problem Research Articles

On approximate vector variational inequalities and vector optimization problem using convexificator

<abstract><p>In the present article, we study a vector optimization problem involving convexificator-based locally Lipschitz approximately convex functions and give some ideas for approximate efficient solutions. In terms of the convexificator, we approximate Stampacchia-Minty type vector variational inequalities and use them to describe an approximately efficient solution to the nonsmooth vector optimization problem. Moreover, we give a numerical example that attests to the credibility of our results.</p></abstract>

Accelerated Continuous-Time Approximate Dynamic Programming via Data-Assisted Hybrid Control

We introduce a new closed-loop architecture for the online solution of approximate optimal control problems in the context of continuous-time systems. Specifically, we introduce the first algorithm that incorporates dynamic momentum in actor-critic structures to control continuous-time dynamic plants with an affine structure in the input. By incorporating dynamic momentum in our algorithm, we are able to accelerate the convergence properties of the closed-loop system, achieving superior transient performance compared to traditional gradient-descent based techniques. In addition, by leveraging the existence of past recorded data with sufficiently rich information properties, we dispense with the persistence of excitation condition traditionally imposed on the regressors of the critic and the actor. Given that our continuous-time momentum-based dynamics also incorporate periodic discrete-time resets that emulate restarting techniques used in the machine learning literature, we leverage tools from hybrid dynamical systems theory to establish asymptotic stability properties for the closed-loop system. We illustrate our results with a numerical example.

The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

Data-Driven Adaptive Network Slicing for Multi-Tenant Networks

Network slicing to support multi-tenancy plays a key role in improving the performance of 5G networks. In this paper, we propose a two time-scale framework for the reservation-based network slicing in the backhaul and Radio Access Network (RAN). In the proposed two time-scale scheme, a subset of network slices is activated via a novel sparse optimization framework in the long time-scale with the goal of maximizing the expected utilities of tenants while in the short time-scale the activated slices are reconfigured according to the time-varying user traffic and channel states. Specifically, using the statistics from users and channels and also considering the expected utility from serving users of a slice and the reconfiguration cost, we formulate a sparse optimization problem to update the configuration of a slice resources such that the maximum isolation of reserved resources is enforced. The formulated optimization problems for long and short time-scales are non-convex and difficult to solve. We use the $\ell_q$-norm, $0<q<1$, and group LASSO regularizations to iteratively find convex approximations of the optimization problems. We propose a Frank-Wolfe algorithm to iteratively solve approximated problems in long time-scales. To cope with the dynamical nature of traffic variations, we propose a fast, distributed algorithm to solve the approximated optimization problems in short time-scales. Simulation results demonstrate the performance of our approaches relative to optimal solutions and the existing state of the art method.

Progressive coherence and spectral norm minimization scheme for measurement matrices in compressed sensing

Compressed sensing is a novel signal sampling theory based on the sparsity of signals. To recover a sparse signal from fewer measurements, it is desired that measurement matrices have low coherence and low spectral norm. For this purpose, a progressive coherence and spectral norm minimization (PCSNM) scheme is proposed by dividing the optimization problem into several ℓ∞-minimization problems with orthogonality constraints. Utilizing the smoothing technique and penalty function method, these non-smooth constrained problems are transformed into unconstrained approximate optimization problems and solved by a gradient-type based alternating minimization approach. Simulation results demonstrate that the proposed scheme outperforms some previous methods, especially in decreasing the spectral norm.

Performance Analysis and Optimization of a UAV-Enabled Two-Way Relaying Network Under FSMH, NC, and PNC Schemes

Unmanned aerial vehicles (UAVs) have played an important role in wireless communications due to the advantages, such as highly controllable mobility in 3-D space, swift deployment, Line-of-Sight (LoS) aerial–ground links, and so on. In this article, we consider a UAV-enabled two-way relaying system, where the UAV relay assists the information exchange between two ground users (GUs) under three different schemes, i.e., four-slot multihopping (FSMH) without network coding (NC), three-slot NC, and two-slot physical NC (PNC). First, the capacity region of each scheme in this relaying system is analyzed. Then, we maximize the system average sum rate by jointly optimizing the time resource allocation, transmission powers of the transceivers, and the UAV trajectory subject to the constraints on UAV mobility and information causality under each scheme. To solve those problems, we propose an iterative algorithm by applying the successive convex approximation and block coordinate descent techniques. Specifically, the time resource allocation, transmission powers, and the UAV trajectory are alternatively optimized in each iteration. In addition, the nonconvex trajectory optimization problem is solved by successively solving an approximate convex optimization problem. To gain more insights, we also investigate the performance of those three schemes with symmetric and asymmetric traffic, respectively, by introducing a new traffic pattern constraint. Numerical results show that the proposed relaying schemes with moving relay can achieve great throughput gains as compared to the conventional scheme with static relay. The three relay schemes also show great performance heterogeneity under different traffic patterns.

Adaptive multi‐surrogate‐based constrained optimization method and its application

AbstractThis article presents an adaptive multi‐surrogate constrained optimization method (AMSCOM) that can automatically determine the appropriate metamodel for each black‐box function in the constrained optimization problem (COP) and concurrently find the optimum. In AMSCOM, each black‐box function is approximated initially by several different types of candidate surrogates. Then, as optimization progresses, the poorly performing candidate surrogates of each black‐box function are gradually eliminated until the appropriate surrogate is found. Meanwhile, as more than one candidate surrogate exists for each unknown function in the optimization process, multiple approximate optimization problems (AOPs) can be constructed, and new samples can be obtained by solving these AOPs. Additionally, we employ the genetic operator and the local‐linear approximation–Voronoi method to generate new samples. To verify the effectiveness and investigate several properties of AMSCOM, the proposed method is tested on 12 benchmark COPs and compared with several single surrogate‐based methods. Furthermore, AMSCOM is compared with several published surrogate‐based constrained optimization methods, and the results further prove the superior performance of AMSCOM. The proposed method is then employed to optimize the shaft‐clinching process of wheel‐hub‐bearing units, and a desirable result is achieved.

Approximate optimal influence over an agent through an uncertain interaction dynamic

An approximate optimal indirect regulation problem is considered for two nonlinear uncertain agents. An influencing agent is tasked with optimally intercepting and directing a roaming agent to a goal location. The roaming agent is not directly controlled by the influencing agent, but instead moves based on some uncertain interaction dynamic. To overcome this challenge, a virtual controller is designed to yield a desired influence on the roaming agent. In addition, an approximate dynamic programming (ADP) strategy is used to develop an approximate optimal solution to the optimal control problem using a computationally efficient function approximation method. Because system uncertainties are considered in both agents, a data-based parameter identification method called integral concurrent learning (ICL) is used to identify uncertain dynamics. A Lyapunov-based stability analysis is performed which proves the closed-loop pursuing and roaming agent systems are uniformly ultimately bounded (UUB). Simulation and experimental results are provided to demonstrate the performance of the developed method.

A branch-and-bound algorithm for the exact optimal experimental design problem

We discuss a generalisation of the approximate optimal experimental design problem, in which the weight of each regression point needs to stay in a closed interval. We work with Kiefer’s optimality criteria which include the well-known D- and A-optimality as special cases. We propose a first-order algorithm for the generalised problem that redistributes the weights of two regression points in each iteration. We develop a branch-and-bound algorithm for exact optimal experimental design problems under Kiefer’s criteria where the subproblems in the search tree are equivalent to the generalized approximate design problem, and therefore, can be solved efficiently by the first-order method. We observe that our branch-and-bound algorithm is favourable to a popular exchange heuristic for certain problem instances.

Adaptation of metaheuristic algorithms to improve training performance of an ESZSL model

Zero-shot learning (ZSL) is a recent promising learning approach that is similar to human vision systems. ZSL essentially allows machines to categorize objects without requiring labeled training data. In principle, ZSL proposes a novel recognition model by specifying merely the attributes of the category. Recently, several sophisticated approaches have been introduced to address the challenges regarding this problem. Embarrassingly simple approach to zeroshot learning (ESZSL) is one of the critical of those approaches that basically proposes a simple but efficient linear code solution. However, the performance of the ESZSL model mainly depends on parameter selection. Metaheuristic algorithms are considered as one the most sophisticated computational intelligence paradigms that allows to approximate optimization problems with high success. This paper addresses this problem by adapting leading metaheuristic algorithms to automatically train the parameters of a linear ESZSL model. The model is statistically validated by performing a series of experiments with benchmark datasets.

Parametric approximate optimal control of uncertain differential game with application to counter terror

Abstract The linear quadratic differential game plays an important role in many fields. It is well known that the saddle point of linear quadratic differential game is given in a feedback form with a solution of Riccati differential equation. However, the control-related Riccati differential equation cannot be solved analytically in many cases. Then the optimal controls may be difficult to be implemented in practice. In order to simplify the forms of optimal controls, in this paper, we investigate a parametric approximate optimal control problem of linear quadratic differential game under uncertain environment. First, we introduce an uncertain linear quadratic differential game model and its analytic optimal controls. Then, an uncertain linear quadratic differential game model with constrained parametric control domain is formulated. Moreover, a parametric approximate optimization method is presented for solving the optimal control parameters. Finally, a counter terror problem is analyzed to show the efficiency of our presented method.

The uncoupled microbial fed-batch fermentation optimization based on state-dependent switched system

In the actual microbial fermentation process, excessive or insufficient substrate can produce inhibitory effects on cells growth. The artificial substrate feeding rules by past experiences have great blindness to keep substrate concentration in a given appropriate range. This paper considers that alkali feed depends on pH value of the solution and glycerol feed depends on glycerol concentration of the solution in the uncoupled microbial fed-batch fermentation process, and establishes a state-dependent switched system in which the flow rates of glycerol and alkali, the number of mode switches, the mode sequence and the switching times are prior unknown. To maximize the yield of target product 1,3-Propanediol (1,3-PD), we formulate a switching optimal control problem with the flow rates of glycerol and alkali, the number of mode switches, the mode sequence and the switching times as decision variables, which is a mixed-integer dynamic programming problem. For solving the mixed-integer dynamic programming problem, the control parametrization technique, the time scaling transformation and the embedded system technology are used to obtain an approximate parameter optimization problem. By using a parallel optimization algorithm, we obtain the optimal control strategies. Under the obtained optimal control strategies, the 1,3-PD yield at the terminal time is increased significantly compared with the previous results.

A direct method for updating mass and stiffness matrices with submatrix constraints

The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let and be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that where M (r) and K (r) are the r × r leading principal submatrices of M and K, respectively. We then consider an optimal approximation problem (OAP): Given n × n symmetric matrices M a and K a . Find such that , where is the solution set of Problem IP-MUP. In this paper, the solvability condition for Problem IP-MUP is established, and the expression of the general solution of Problem IP-MUP is derived. Also, we show that the optimal approximation solution is unique and derive an explicit formula for it.

Metrics, Quantization and Registration in Varifold Spaces

This paper is concerned with the theory and applications of varifolds to the representation, approximation and diffeomorphic registration of shapes. One of its purpose is to synthesize and extend several prior works which, so far, have made use of this framework mainly in the context of submanifold comparison and matching. In this work, we instead consider deformation models acting on general varifold spaces, which allow to formulate and tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. We study in detail the construction of kernel metrics on varifold spaces and the resulting topological properties of those metrics and then propose a mathematical model for diffeomorphic registration of varifolds under a specific group action which we formulate in the framework of optimal control theory. A second important part of the paper focuses on the discrete aspects. Specifically, we address the problem of optimal finite approximations (quantization) for those metrics and show a $$\varGamma $$ -convergence property for the corresponding registration functionals. Finally, we develop numerical pipelines for quantization and registration before showing a few preliminary results for one- and two-dimensional varifolds.

Moreau Envelope of Supremum Functions with Applications to Infinite and Stochastic Programming

In this paper, we investigate the Moreau envelope of the supremum of a family of convex, proper, and lower semicontinuous functions. Under mild assumptions, we prove that the Moreau envelope of a supremum is the supremum of Moreau envelopes, which allows us to approximate possibly nonsmooth supremum functions by smooth functions that are also the suprema of functions. Consequently, we propose and study approximated optimization problems from infinite and stochastic programming for which we obtain zero-duality gap results and optimality conditions without the verification of constraint qualification conditions.

Natural evolution strategies and variational Monte Carlo

A notion of quantum natural evolution strategies is introduced, which provides a geometric synthesis of a number of known quantum/classical algorithms for performing classical black-box optimization. The recent work of Gomes et al (2019 arXiv:1910.10675) on heuristic combinatorial optimization using neural quantum states is pedagogically reviewed in this context, emphasizing the connection with natural evolution strategies (NES). The algorithmic framework is illustrated for approximate combinatorial optimization problems, and a systematic strategy is found for improving the approximation ratios. In particular, it is found that NES can achieve approximation ratios competitive with widely used heuristic algorithms for Max-Cut, at the expense of increased computation time.

Submatrix constrained left and right inverse eigenvalue problem for centrosymmetric matrices

ABSTRACT In this article, we will find centrosymmetric matrix solutions A of the left and right inverse eigenvalue problem under a submatrix constraint, where is also a centrosymmetric matrix. In other words, expand the system (matrix) A from the centre subsystem (submatrix) satisfying the matrix constraint, where A and are both centrosymmetric matrices. Using the similar structure of A and , we discuss the sufficient and necessary conditions for the left and right inverse eigenvalue problem having solutions, and give the expression for its general solution. Then, we discuss its optimal approximation problem and gain the expression of its solution. Last, we provide a feasible algorithm for computing the unique solution to its optimal approximation problem, which is proved by some numerical examples.

Resource Allocation for IRS-Assisted Full-Duplex Cognitive Radio Systems

In this article, we investigate the resource allocation design for intelligent reflecting surface (IRS)-assisted full-duplex (FD) cognitive radio systems. In particular, a secondary network employs an FD base station (BS) for serving multiple half-duplex downlink (DL) and uplink (UL) users simultaneously. An IRS is deployed to enhance the performance of the secondary network while helping to mitigate the interference caused to the primary users (PUs). The DL transmit beamforming vectors and the UL receive beamforming vectors at the FD BS, the transmit power of the UL users, and the phase shift matrix at the IRS are jointly optimized for maximization of the total spectral efficiency of the secondary system. The design task is formulated as a non-convex optimization problem taking into account the imperfect knowledge of the PUs’ channel state information (CSI) and their maximum interference tolerance. Since the maximum interference tolerance constraint is intractable, we apply a safe approximation to transform it into a convex constraint. To efficiently handle the resulting approximated optimization problem, which is still non-convex, we develop an iterative block coordinate descent (BCD)-based algorithm. This algorithm exploits semidefinite relaxation, a penalty method, and successive convex approximation and is guaranteed to converge to a stationary point of the approximated optimization problem. Our simulation results do not only reveal that the proposed scheme yields a substantially higher system spectral efficiency for the secondary system than several baseline schemes, but also confirm its robustness against CSI uncertainty. Besides, our results illustrate the tremendous potential of IRS for managing the various types of interference arising in FD cognitive radio networks.

Time Allocation and Optimization in Time-Reversal Wireless Powered Communication Networks

This paper studies optimal resource allocation in the wireless powered communication networks (WPCN) combined with time reversal (TR) in which one hybrid access point (H-AP) broadcasts constant wireless energy to a set of distributed users in the downlink (DL) and receives information from the users via space division multiple access (SDMA) in the uplink (UL). Inevitable interferences will occur when users transmit information in the UL simultaneously and the special space-time focusing of TR is used to suppress the interferences. An efficient protocol is proposed to support wireless energy transfer (WET) and TR in the DL and wireless information transmission in the UL for the proposed TR-WPCN. We optimize the time allocations to the H-AP for DL WET, DL TR, and UL WIT to maximize the sum throughput. Due to the nonconvexity of the studied optimization problem, we optimize variables successively, where the nonconvex optimization problem is transformed into the convex optimization problem. The approximate convex optimization problem can then be solved iteratively combined with the dichotomy method. Simulation results show that the proposed scheme can effectively suppress interferences and improve system performance.

Integral Reinforcement Learning-Based Multi-Robot Minimum Time-Energy Path Planning Subject to Collision Avoidance and Unknown Environmental Disturbances

In this letter, we study the online multi-robot minimum time-energy path planning problem subject to collision avoidance and input constraints in an unknown environment. We develop an online adaptive solution for the problem using integral reinforcement learning (IRL). This is achieved through transforming the finite-horizon minimum time-energy problem with input constraints to an approximate infinite-horizon optimal control problem. To achieve collision avoidance, we incorporate artificial potential fields into the approximate cost function. We develop an IRL-based optimal control strategy and prove its convergence. The theoretical results are verified through simulation studies.