Exact Penalty Research Articles

Beamforming Design in Multiple-Input-Multiple-Output Symbiotic Radio Backscatter Systems

Symbiotic radio (SR) backscatter is a possible low-power wireless communication technique for future Internet of Things devices. In this letter, a multiple-input-multiple-output (MIMO) SR backscatter system is proposed, where the secondary multi-antenna transmission from the backscatter device (BD) to the receiver is riding on the primary multi-antenna transmission from the transmitter to the receiver. We investigate the beamforming design optimization problem which maximizes the sum capacity of primary and secondary transmissions under the capacity constraint of secondary transmission. In the MIMO SR backscatter system, each antenna of the SR BD reflects its received ambient radio frequency signals from all the transmitting antennas of the transmitter, which causes that the globally optimal solution is difficult to obtain. Here, we derive an algorithm to obtain the capacity upper bound. Furthermore, considering both primary and secondary transmissions, we propose an exact penalty method based locally optimal solution. Simulation results display that the proposed exact penalty method based locally optimal solution achieves the sum capacity which is close to the upper bound.

Trajectory planning and automatic obstacle avoidance algorithm for unmanned surface vehicle based on exact penalty function

Trajectory planning and automatic obstacle avoidance algorithm for unmanned surface vehicle based on exact penalty function

EXPEDIS: An exact penalty method over discrete sets

We address the problem of minimizing a quadratic function subject to linear constraints over binary variables. We introduce the exact solution method called EXPEDISwhere the constrained problem is transformed into a max-cut instance, and then the whole machinery available for max-cut can be used to solve the transformed problem. We derive the theory in order to find a transformation in the spirit of an exact penalty method; however, we are only interested in exactness over the set of binary variables. In order to compute the maximum cut we use the solver BiqMac. Numerical results show that this algorithm can be successfully applied on various classes of problems.

Artificial Noise-Aided Secure Relay Communication With Unknown Channel Knowledge of Eavesdropper

In this article, a new relay-aided secure communication system is investigated, where a transmitter sends signals to a destination via an amplify-and-forward (AF) relay in the presence of an eavesdropper. We consider a general system configuration, where the source, relay, destination, and eavesdropper are all equipped with multiple antennas. In the practical scenarios of unknown eavesdropper's channel state information (CSI) and uncertainty of the eavesdropper's location, we aim to maximize the expected value of the system secrecy rate over the presumed distribution of the eavesdropper's channels, by exploiting the artificial noise (AN) transmitted by the source and relay nodes. The system design issue is formulated as a nonconvex stochastic optimization problem with a source transmission power constraint and a nonconvex relay transmission power constraint. A novel computational method is proposed to solve this challenging problem. The new method is developed based on an exact penalty function method together with a parallel stochastic decomposition algorithm. Numerical simulations are performed to study the effectiveness of the proposed scheme at various locations of the eavesdropper. Simulation results show that for most cases, secure communication can be achieved without the CSI knowledge of eavesdropper's channels, and the achievable secrecy rate follows the trend of a benchmark system where the eavesdropper's full CSI is available. In particular, the achievable system secrecy rate increases with the number of antennas at the legitimate users. Moreover, the optimal power allocated for the transmission of the AN increases with the system signal-to-noise ratio. The proposed computational method achieves a higher system secrecy rate than a conventional penalty function based approach.

A difference-of-convex programming approach with parallel branch-and-bound for sentence compression via a hybrid extractive model

Sentence compression is an important problem in natural language processing with wide applications in text summarization, search engine and human-AI interaction system etc. In this paper, we design a hybrid extractive sentence compression model combining a probability language model and a parse tree language model for compressing sentences by guaranteeing the syntax correctness of the compression results. Our compression model is formulated as an integer linear programming problem, which can be rewritten as a Difference-of-Convex (DC) programming problem based on the exact penalty technique. We use a well-known efficient DC algorithm -- DCA to handle the penalized problem for local optimal solutions. Then a hybrid global optimization algorithm combining DCA with a parallel branch-and-bound framework, namely PDCABB, is used for finding global optimal solutions. Numerical results demonstrate that our sentence compression model can provide excellent compression results evaluated by F-score, and indicate that PDCABB is a promising algorithm for solving our sentence compression model.

A Bregman Forward-Backward Linesearch Algorithm for Nonconvex Composite Optimization: Superlinear Convergence to Nonisolated Local Minima

We introduce Bella, a locally superlinearly convergent Bregman forward backward splitting method for minimizing the sum of two nonconvex functions, one of which satisfying a relative smoothness condition and the other one possibly nonsmooth. A key tool of our methodology is the Bregman forward-backward envelope (BFBE), an exact and continuous penalty function with favorable first- and second-order properties, and enjoying a nonlinear error bound when the objective function satisfies a Lojasiewicz-type property. The proposed algorithm is of linesearch type over the BFBE along candidate update directions, and converges subsequentially to stationary points, globally under a KL condition, and owing to the given nonlinear error bound can attain superlinear convergence rates even when the limit point is a nonisolated minimum, provided the directions are suitably selected.

A New Unified Path to Smoothing Nonsmooth Exact Penalty Function for the Constrained Optimization

We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization problem, and discuss the convergence of the algorithm under mild conditions.

Finite-Alphabet Symbol-Level Multiuser Precoding for Massive MU-MIMO Downlink

We propose a finite-alphabet symbol-level precoding technique for massive multiuser multiple-input multiple-output (MU-MIMO) downlink systems which are equipped with finite-resolution digital-to-analog converters (DACs) of any precision. Using the idea of constructive interference (CI), we adopt a max-min fair design criterion which aims to maximize the minimum instantaneous received signal-to-noise ratio (SNR) among the user equipments (UEs) while ensuring a CI constraint for each UE under the restriction that the output of the precoder is a vector with finite-alphabet discrete elements. Due to this latter constraint, the design problem is an NP-hard quadratic program with discrete variables, and hence, is difficult to solve. In this paper, we tackle this difficulty by reformulating the problem in several steps into an equivalent continuous-domain biconvex form, including equivalent representations for discrete and binary constraints. Our final biconvex reformulation is obtained via an exact penalty approach and can efficiently be solved using a standard cyclic block coordinate descent algorithm. We evaluate the performance of the proposed finite-alphabet precoding design for DACs with different resolutions, where it is shown that employing low-resolution DACs can lead to higher power efficiencies. In particular, we focus on a setup with one-bit DACs and show through simulation results that compared to the existing schemes, the proposed design can achieve SNR gains of up to 2 dB. We further provide analytic and numerical analyses of complexity and show that our proposed algorithm is computationally efficient as it typically needs only a few tens of iterations to converge.

Exact penalty functions in the problem of choosing the optimal wholesale order in the face of rapid fluctuations in demand

The current article discusses a different situation in the market, when there is a rush demand for a new product followed by a sharp drop in demand. The trading company uses the following scheme for the wholesale order of goods. The ordered product is divided into two parts, and the first batch of goods arrives immediately, it is sold over a certain period of time (0, Z,]. The second batch of goods is delivered at the time 7’, but at the time interval [0, 7]. This batch is sold at a discount and is completely sold out. Time T' corresponds to the end of the sale of the entire product. The time points 7), T' are selected by the trading firm from the condition of maximizing income. The need to consider such a wholesale order scheme is related to the fact that, firstly, the warehouses of the trading firms have limited capacity and cannot accommodate all the ordered goods, and secondly, a manufacturer may not offer the entire ordered batch of goods, since not all goods can be produced at the initial (zero) point of time immediately after receiving the order. At the time 71, the trading company completely sells the first batch of goods and receives financial resources, part of which is paid to the manufacturer. At the moment of time Т', Ме сотр/ефе sale of all purchased goods is completed. The choice of time points 7; and T' allow to determine the volume of the first batch of ordered goods and the total volume of product ordered from the manufacturer. In the article, a mathematical model is proposed that makes it possible to choose the optimal ordering strategy for a trading company in the conditions of excessive growth of demand for the new product in time 71,72 at some unknown point in time and т. Е [73,74], and the subsequent sharp drop in demand in the period of time [73, 74] due to the saturation of the market with a new product. Four possible variants of optimization problems are considered. A method of exact penalty function is suggested, which allows one to find their solutions.

Design and Application of Intelligence Algorithms in Continuous Fermentation of Glycerol

The bioconversion of 1,3-propanediol from glycerol by Klebsiella pneumoniae can be described by a nonlinear dynamic system. Some work has been done on the identification and optimization of the system, in which the dilution rate of glycerol is considered as a constant. However, the demand of glycerol may vary at different fermentation stages, it is reasonable to view the glycerol metabolic system with dilution rate varying with time. In this paper, we model the glycerol metabolic process as a fourteen-dimensional nonlinear dynamical system, where the dilution rate is considered varying with time. Then an optimal discrete-valued control problem for maximizing the average concentration of 1,3-propanediol in the fermentation process is established. To solve the optimization problem, auxiliary control and an exact penalty function are used to convert this problem into a large-scale parameter optimization problem. For better balancing local and global search ability, a competitive particle swarm algorithm with time-varying control factors is proposed which is proved to be faster and more stable than the traditional competitive particle swarm algorithm. Numerical experiments are conducted to show the rationality, effectiveness and applicability of the method proposed.

On nonconvex optimal control problems

The paper addresses the general optimal control (OC) problem with inequality constraints and a cost functional of Bolza given by d.c. functions with respect to the state in the terminal and integrand parts of the functionals. First, we reduce the original OC problem with inequality constraints to the one without constraints with the help of the Exact Penalization Theory. Further, we show that the auxiliary (penalized) problem also possesses the state-DC-structure. Employing this property, we develop the new Global Optimality Conditions (GOCs) and discuss some its features allowing to construct the new schemes of local and global searchers. Finally we elucidate the relations of the GOCs to the classical OC theory, in particular, to the Pontryagin’s maximum principle.

A class of smooth exact penalty function methods for optimization problems with orthogonality constraints

ABSTRACT Updating the augmented Lagrangian multiplier by closed-form expression yields efficient first-order infeasible approach for optimization problems with orthogonality constraints. Hence, parallelization becomes tractable in solving this type of problems. Inspired by this closed-form updating scheme, we propose a novel penalty function with compact convex constraints (PenC). We show that PenC can act as an exact penalty model which shares the same global minimizers as the original problem with orthogonality constraints. Based on PenC, we first propose a first-order algorithm called PenCF and establish its global convergence and local linear convergence rate under some mild assumptions. For the case that the computation and storage of Hessian is achievable, and we pursue high precision solution and fast local convergence rate, a second-order approach called PenCS is proposed for solving PenC. To avoid expensive calculation or solving a hard subproblem in computing the Newton step, we propose a new strategy to do it approximately which still leads to quadratic convergence locally. Moreover, the main iterations of both PenCF and PenCS are orthonormalization-free and hence parallelizable. Numerical experiments illustrate that PenCF is comparable with the existing first-order methods. Furthermore, PenCS shows its stability and high efficiency in obtaining high precision solution comparing with the existing second-order methods.

Distributed Discrete-Time Algorithms for Convex Optimization With General Local Constraints on Weight-Unbalanced Digraph

This article investigates the convex optimization problem with general local constraints including $N$ equality constraints, $N$ inequality constraints, and $N$ closed convex set constraints. Especially, the objective function of the considered problem is the sum of $N$ convex functions, which are seen as local objective functions. First, for the distributed setting, a weight-unbalanced digraph with $N$ nodes is introduced to describe the communication topology. Clearly, in the distributed setting, each node is responsible for a local objective function accompanied with an equality constraint, an inequality constraint, and a closed convex set constraint, which are viewed as local constraints. The aim of this article is to solve the considered convex optimization problem in a distributed manner, i.e., to lead the states of all nodes to a common optimal solution of the considered problem under the condition that each node only knows its own local objective function, and local constraints, and the information of its neighbors’ states. Towards this end, by resorting to the exact penalty function method and the gradient descent-like method, two distinct distributed discrete-time algorithms are developed for two different cases, respectively. Furthermore, the convergence properties of the designed algorithms are rigorously analyzed under some common and standard assumptions, and the convergence rates are described in detail. Finally, simulation results are provided to verify the theoretical results.

Multidisciplinary collaborative optimization based on relaxation method for solving complex problems

The purpose of the present work is to improve the performance of the standard collaborative optimization (CO) approach based on an existing dynamic relaxation method. This approach may be weakened by starting design points. First, a New Relaxation (NR) method is proposed to solve the difficulties in convergence and low accuracy of CO. The new method is based on the existing dynamic relaxation method and it is achieved by changing the system-level consistency equality constraints into relaxation inequality constraints. Then, a Modified Collaborative Optimization (MCO) approach is proposed to eliminate the impact of the information inconsistency between the system-level and the discipline-level on the feasibility of optimal solutions. In the MCO approach, the impact of the inconsistency is treated by transforming the discipline-level constrained optimization problems into an unconstrained optimization problem using an exact penalty function. Based on the NR method, the performance of the MCO approach carried out by solving two multidisciplinary optimization problems. The obtained results show that the MCO approach has improved the convergence of CO significantly. These results prove that the present MCO succeeds in getting feasible solutions while the CO fails to provide feasible solutions with the used starting design points.

Exact Penalization of Generalized Nash Equilibrium Problems

This paper presents an exact penalization theory of the generalized Nash equilibrium problem (GNEP) that has its origin from the renowned Arrow–Debreu general economic equilibrium model. Whereas the latter model is the foundation of much of mathematical economics, the GNEP provides a mathematical model of multiagent noncooperative competition that has found many contemporary applications in diverse engineering domains. The most salient feature of the GNEP that distinguishes it from a standard noncooperative (Nash) game is that each player’s optimization problem contains constraints that couple all players’ decision variables. Extending results for stand-alone optimization problems, the penalization theory aims to convert the GNEP into a game of the standard kind without the coupled constraints, which is known to be more readily amenable to solution methods and analysis. Starting with an illustrative example to motivate the development, this paper focuses on two kinds of coupled constraints, shared (i.e., common) and finitely representable. Constraint residual functions and the associated error bound theory play an important role throughout the development.

Modelling and optimal state-delay control in microbial batch process

In this paper, we consider optimal control problem involving a time-varying state-delay system arising in 1,3-propanediol microbial batch process. The dynamic system in this problem includes unknown time-varying delay function and unknown kinetic parameters. To optimally determine the unknown delay function and unknown kinetic parameters in the system, the weighted least-squares error between the computed values and experimental data is minimized subject to path constraints. By parameterizing the delay function with piecewise quadratic basis functions, the optimal state-delay control problem is approximated by a sequence of parameter optimization problems. Furthermore, an exact penalty method is utilized to transform these parameter optimization problems into the ones only with box constraints. On this basis, a modified differential evolution algorithm is developed to solve the resulting optimization problems. Finally, numerical results are presented to verify the effectiveness of the developed solution approach.

A Level-Set Approach for Stochastic Optimal Control Problems Under Controlled-Loss Constraints

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton–Jacobi–Bellman equation usually calls for strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in the absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then solve the latter by a level-set approach. With this approach, state constraints are managed through an exact penalization technique.

Smoothing Partially Exact Penalty Function of Biconvex Programming

In this paper, a smoothing partial exact penalty function of biconvex programming is studied. First, concepts of partial KKT point, partial optimum point, partial KKT condition, partial Slater constraint qualification and partial exactness are defined for biconvex programming. It is proved that the partial KKT point is equal to the partial optimum point under the condition of partial Slater constraint qualification and the penalty function of biconvex programming is partially exact if partial KKT condition holds. We prove the error bounds properties between smoothing penalty function and penalty function of biconvex programming when the partial KKT condition holds, as well as the error bounds between objective value of a partial optimum point of smoothing penalty function problem and its [Formula: see text]-feasible solution. So, a partial optimum point of the smoothing penalty function optimization problem is an approximately partial optimum point of biconvex programming. Second, based on the smoothing penalty function, two algorithms are presented for finding a partial optimum or approximate [Formula: see text]-feasible solution to an inequality constrained biconvex optimization and their convergence is proved under some conditions. Finally, numerical experiments show that a satisfactory approximate solution can be obtained by the proposed algorithm.

Initialization-Free Distributed Fixed-Time Convergent Algorithms for Optimal Resource Allocation

This article considers the distributed fixed-time resource allocation problem subject to global equality and local inequality constraints. In the case that the optimal solutions are strictly feasible, a projection-gradient-based continuous-time algorithm is first proposed to minimize a team of cost functions in the fixed time. As some optimal solutions locate on the boundary of local constraints, we further provide an alternative suboptimal solution based on the <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-exact penalty function method. Different from the existing distributed optimal allocation results, the (sub)optimal solutions can be obtained in the fixed time and the optimization protocols can be implemented in an initialization-free manner. Thus, the convergence time can be offline preassigned and the initial states can be chosen arbitrarily such that a better robust performance is achieved for the new algorithms. Case studies of the widely discussed economic dispatch problems are performed to validate the effectiveness of the obtained results.

Multi-objective optimisation of nonlinear switched systems in uncoupled fed-batch fermentation

Maximising yield and minimising time are two top priorities for biotechnological industry. In this paper, we model the 1,3-PD uncoupled fed-batch fermentation process with a 10-dimensional nonlinear switched system. By taking both maximising 1,3-PD concentration and minimising the terminal time of fermentation as the objectives, we propose a multi-objective optimisation problem with the flow rates of glycerol and alkali, the switched times among four modes and the mode sequence as the decision variables. To solve the multi-objective optimisation problem, we use the convex weighted sum to transcribe the multi-objective optimisation problem into a sequence of single-objective optimisation problems. Then the time-scaling transformation, the binary relaxation approach and the exact penalty method are applied to solve these single-objective optimisation problems. The numerical results show that, by optimising the flow rates of glycerol and alkali, the switched times among four modes and the mode sequence, the optimum fed-batch time and the higher 1,3-PD concentration can be obtained. The obtained numerical results can provide reference for the actual fermentation experiment.