Nonconvex Constrained Optimization Problems Research Articles

A shifted hyperbolic augmented Lagrangian-based artificial fish two-swarm algorithm with guaranteed convergence for constrained global optimization

This article presents a shifted hyperbolic penalty function and proposes an augmented Lagrangian-based algorithm for non-convex constrained global optimization problems. Convergence to an -global minimizer is proved. At each iteration k, the algorithm requires the -global minimization of a bound constrained optimization subproblem, where . The subproblems are solved by a stochastic population-based metaheuristic that relies on the artificial fish swarm paradigm and a two-swarm strategy. To enhance the speed of convergence, the algorithm invokes the Nelder–Mead local search with a dynamically defined probability. Numerical experiments with benchmark functions and engineering design problems are presented. The results show that the proposed shifted hyperbolic augmented Lagrangian compares favorably with other deterministic and stochastic penalty-based methods.

Optimal back-off point determination and controller weight selection for multivariate systems under finite-horizon control

Plant economic performance is most often related to the operating point, specifically the mean values of the process variables; meanwhile, most existing performance assessment techniques involve examining the variances or covariances of the controlled variables. A combined approach is to determine the appropriate trade-off between variances of different process variables in order to operate the plant at the point that provides maximum economic benefit while satisfying the operating constraints. This problem is referred to as the minimum backed-off operating point selection, and previous works have formulated it as a non-convex constrained optimization problem. In the current work, a new technique is introduced that can provide the optimal plant operating point. Additionally, this method provides the weights for a finite horizon controller that results in the optimal trade-off in process variable variances that will allow satisfaction of the operating constraints at the optimal operating point. In this method, the plant and disturbance models for the given process are used to generate data representing possible trade-offs between process variable standard deviations. Employing a piecewise linear regression to describe the sample points of this standard deviations data allows for the operating point selection problem to be solved as a small number of linear programs. The advantages of this approach are demonstrated through the use of mathematical and simulation case studies.

Exchange market algorithm based optimum reactive power dispatch

In this paper, an exchange market algorithm (EMA) approach is applied to solve highly non-linear power system optimal reactive power dispatch (ORPD) problems. ORPD is most vital optimization problems in power system study and are usually devised as optimal power flow (OPF) problem. The problem is formulated as nonlinear, non-convex constrained optimization problem with the presence of both continuous and discrete control variables. The EMA searches for optimal solution via two main phases; namely, balanced market and oscillation market. Each of the phases comprises of both exploration and exploitation, which makes the algorithm unique. This uniqueness of EMA is exploited in this paper to solve various vital objectives associated with ORPD problems. Programs are developed in MATLAB and tested on standard IEEE 30 and IEEE 118 bus systems. The results obtained using EMA are compared with other contemporary methods in the literature. Simulation results demonstrate the superiority of EMA in terms of its computational efficiency and robustness. Consumed function evaluation for each case study is mentioned in the convergence plot itself for better clarity. Parametric study is also performed on different case studies to obtain the suitable values of tuneable parameters.

Iterated non-linear model predictive control based on tubes and contractive constraints

This paper presents a predictive control algorithm for non-linear systems based on successive linearizations of the non-linear dynamic around a given trajectory. A linear time varying model is obtained and the non-convex constrained optimization problem is transformed into a sequence of locally convex ones. The robustness of the proposed algorithm is addressed adding a convex contractive constraint. To account for linearization errors and to obtain more accurate results an inner iteration loop is added to the algorithm. A simple methodology to obtain an outer bounding-tube for state trajectories is also presented. The convergence of the iterative process and the stability of the closed-loop system are analyzed. The simulation results show the effectiveness of the proposed algorithm in controlling a quadcopter type unmanned aerial vehicle.

Energy-Efficient Cooperative Tfor Simultaneous Wireless Information and Power Transfer in Clustered Wireless Sensor Networks

This paper considers applying simultaneous wireless information and power transfer (SWIPT) technique to cooperative clustered wireless sensor networks, where energy-constrained relay nodes harvest the ambient radio-frequency (RF) signal and use the harvested energy to forward the packets from sources to destinations. To this end, we first formulate the energy-efficient cooperative transmission (eCotrans) problem for SWIPT in clustered wireless sensor networks as a non-convex constrained optimization problem. Then, by exploiting fractional programming and dual decomposition, we develop a distributed iteration algorithm for power allocation, power splitting and relay selection to solve the non-convex optimization problem. We find that power splitting ratio plays an imperative role in relay selection. Our simulation results illustrate that the proposed algorithm can converge within a few iterations and the numerical analysis provides practical insights into the effect of various system parameters, such as the number of relay nodes, the inter-cluster distance and the maximum transmission power allowance, on energy efficiency and average harvested power.

Reduced-Complexity Algorithms for Indoor Map-Aware Localization Systems

The knowledge of environmental maps (i.e., map-awareness) can appreciably improve the accuracy of optimal methods for position estimation in indoor scenarios. This improvement, however, is achieved at the price of a significant complexity increase with respect to the case of map-unawareness, specially for large maps. This is mainly due to the fact that optimal map-aware estimation algorithms require integrating highly nonlinear functions or solving nonlinear and nonconvex constrained optimization problems. In this paper, various techniques for reducing the complexity of such estimators are developed. In particular, two novel strategies for restricting the search domain of map-aware position estimators are developed and the exploitation of state-of-the-art numerical integration and optimization methods is investigated; this leads to the development of a new family of suboptimal map-aware localization algorithms. Our numerical and experimental results evidence that the accuracy of these algorithms is very close to that offered by their optimal counterparts, despite their significantly lower computational complexity.

A novel neural network based on NCP function for solving constrained nonconvex optimization problems

This article presents a novel neural network (NN) based on NCP function for solving nonconvex nonlinear optimization (NCNO) problem subject to nonlinear inequality constraints. We first apply the p‐power convexification of the Lagrangian function in the NCNO problem. The proposed NN is a gradient model which is constructed by an NCP function and an unconstrained minimization problem. The main feature of this NN is that its equilibrium point coincides with the optimal solution of the original problem. Under a proper assumption and utilizing a suitable Lyapunov function, it is shown that the proposed NN is Lyapunov stable and convergent to an exact optimal solution of the original problem. Finally, simulation results on two numerical examples and two practical examples are given to show the effectiveness and applicability of the proposed NN. © 2015 Wiley Periodicals, Inc. Complexity 21: 130–141, 2016

Canonical duality for solving general nonconvex constrained problems

This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed.

A Nonmonotone Filter SQP Method: Local Convergence and Numerical Results

The work by Gould, Loh, and Robinson [SIAM J. Optim., 24 (2014), pp. 175--209] established global convergence of a new filter line search method for finding local first-order solutions to nonlinear and nonconvex constrained optimization problems. A key contribution of that work was that the search direction was computed using the same procedure during every iteration from subproblems that were always feasible and computationally tractable. This contrasts previous filter methods that require a separate restoration phase based on subproblems solely designed to reduce infeasibility. In this paper, we present a nonmonotone variant of our previous algorithm that inherits the previously established global convergence property. In addition, we establish local superlinear convergence of the iterates and provide the results of numerical experiments. The numerical tests validate our method and highlight an interesting numerical trade-off between accepting more (on average lower quality) steps versus fewer (on avera...

Robust structured controllers for piezoelectric microactuators

This paper introduces a new control scheme which incorporates the concept of shaping filter together with the use of the ν-gap metric and the robust design of a structured controller. The main motivation in doing this is related to the development of efficient control laws for small size actuators. Designing a structured controller is known to be computationally intractable via the traditional H∞ method. This is mainly due to the non-convexity of the resulting control problem which is of fixed order or structure type. To solve this kind of control problem easily and directly, without using any complicated mathematical manipulations and without using too many “user defined” parameters, we utilize the heuristic Kalman algorithm (HKA) for the resolution of the underlying constrained non-convex optimization problem. The experimental results validate the proposed technique and demonstrate its convenience for the development of fast and precise positioning systems.

A filter-based artificial fish swarm algorithm for constrained global optimization: theoretical and practical issues

This paper presents a filter-based artificial fish swarm algorithm for solving nonconvex constrained global optimization problems. Convergence to an $$\varepsilon $$ -global minimizer is guaranteed. At each iteration $$k$$ , the algorithm requires a $$(\rho ^{(k)},\varepsilon ^{(k)})$$ -global minimizer of a bound constrained bi-objective subproblem, where as $$k\rightarrow \infty $$ , $$\rho ^{(k)}\rightarrow 0$$ gives the constraint violation tolerance and $$\varepsilon ^{(k)} \rightarrow \varepsilon $$ is the error bound defining the accuracy required for the solution. The subproblems are solved by a population-based heuristic known as artificial fish swarm algorithm. Each subproblem relies on the approximate solution of the previous one, randomly generated new points to explore the search space for a global solution, and the filter methodology to accept non-dominated trial points. Convergence to a $$(\rho ^{(k)},\varepsilon ^{(k)})$$ -global minimizer with probability one is guaranteed by probability theory. Preliminary numerical experiments show that the algorithm is very competitive when compared with known deterministic and stochastic methods.

Neural network for nonsmooth, nonconvex constrained minimization via smooth approximation.

A neural network based on smoothing approximation is presented for a class of nonsmooth, nonconvex constrained optimization problems, where the objective function is nonsmooth and nonconvex, the equality constraint functions are linear and the inequality constraint functions are nonsmooth, convex. This approach can find a Clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation. The global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded. Specially, the proposed network does not require: 1) the initial point to be feasible; 2) a prior penalty parameter to be chosen exactly; 3) a differential inclusion to be solved. Numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network.

Error Bounds and Finite Termination for Constrained Optimization Problems

We present a global error bound for the projected gradient of nonconvex constrained optimization problems and a local error bound for the distance from a feasible solution to the optimal solution set of convex constrained optimization problems, by using the merit function involved in the sequential quadratic programming (SQP) method. For the solution sets (stationary points set andKKTpoints set) of nonconvex constrained optimization problems, we establish the definitions of generalized nondegeneration and generalized weak sharp minima. Based on the above, the necessary and sufficient conditions for a feasible solution of the nonconvex constrained optimization problems to terminate finitely at the two solutions are given, respectively. Accordingly, the results in this paper improve and popularize existing results known in the literature. Further, we utilize the global error bound for the projected gradient with the merit function being computed easily to describe these necessary and sufficient conditions.

A smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.

An Improved Optimal Linear Weighted Cooperative Spectrum Sensing Algorithm for Cognitive Radio Sensor Networks

In order to improve the sensing accuracy of the Cognitive Radio Sensor Networks and reduce the interference to the primary user, this paper proposes an improved optimal linear weighted cooperative spectrum sensing scheme on the assumption that the report channel is not ideal. Through mathematical modeling, the spectrum sensing problem is ultimately converted into a constrained nonconvex optimization problem, and the chaotic harmony search (CHS) algorithm is to be used to find the optimal weighting vector value. The simulation results show that the proposed linear cooperative spectrum detection scheme based on the CHS algorithm has better performance than HS, SFLA, EGC, MRC, and MDC algorithm. In addition, the influence of local noise power, report channel noise power, and report channel gain on the performance of the algorithm is analyzed by simulation. The results show that local noise power has greater impact on the sensing performance.

Extended radial epiderivatives of non-convex vector-valued maps and parametric quasiconvex programming

In this paper, we consider extended vector-valued mappings defined on a normed linear space. Based on the recent semicontinuous regularizations related to hypographical and/or epigraphical profile mappings of the considered function introduced, we define semicontinuous radial epiderivatives. We, then, demonstrate that the properties of these epiderivatives amount to properties of hypographical and/or epigraphical profile mappings of the corresponding difference quotient of the underlying function, which simplify fairly well the proofs in the radial epiderivative formulaes. In particular, we stress the impact of semicontinuity, hence, we characterize with new arguments the radial epiderivatives in terms of the suprema and/or infima of the interiorly radial cone of the hypograph and/or epigraph of the considered function. Finally, we obtain optimality conditions for general non-convex constrained vector optimization problems. We apply thereafter the obtained pattern to a parametric quasiconvex programming problem for which we derive necessary and sufficient optimality conditions that are not sensitive to perturbation at the nominal level, yielding henceforth more – and strong at least under asymptotically regular constraints – information than the recent stability results obtained under additional conditions on the regularity of the normal cone to the adjusted sublevel sets of the underlying function.

Cross Layer Optimization for Lifetime Maximization in Wireless Sensor Network Based on Particle Swarm Optimization

Network lifetime is a critical metric in the design of energy-constrained wireless sensor networks. In this paper, we consider the joint cross layer optimization of the physical layer, medium access control layer and routing layer to maximize network lifetime of a multi-sources and single-sink wireless sensor network with energy constraints. We focus on synchronous small-scale sensor network with interference-free link scheduling and practical MPSK link transmission scheme. As the network lifetime maximization problem is a constrained non-convex optimization problem that is difficult to be solved, and the particle swarm optimization algorithm is a good intelligent algorithm, we employ it to solve the problem mentioned above effectively in this paper. The penalty function technique is brought in to work out the constrained optimization problem by converting it to an unconstrained optimization problem. Simulation results show the effectiveness of the proposed algorithm in energy saving and network lifetime maximization, and the particle swarm optimization can solve the network lifetime maximization problem fast and efficiently.

Bi-orthogonal rational discrete wavelet transform with multiple regularity orders and application experiments

This paper proposes an approach for designing a general two-band, FIR, critically sampled, rational rate filter bank (RFB) with perfect reconstruction (PR) and regularity properties. Designs obtained from this approach, when iterated, lead to rational discrete wavelet transforms (RADWTs) with adjustable dilation factor. The RFB design is based on solving a non-convex constrained optimization problem in which the non-linear constraints arise from the perfect reconstruction conditions. An iterative algorithm is used to solve the optimization problem through solving a simplified convex quadratic problem with linear constraints at each iteration step. Some examples are provided to demonstrate the use of bi-orthogonal RADWTs in applications such as signal separation which benefit from decompositions with suitably chosen dilation factors.

Joint power allocation for OFDM system with cooperation at the transmitters

It is known that traditional water-filling (WF) provides a closed form solution for capacity maximization in orthogonal frequency division multiplex (OFDM) system. The solution is derived from a maximum mutual information argument with a single transmitter. Motivated by the novel technology of cooperative communication, we consider a new power allocation problem for OFDM systems with two cooperative transmitters, where each transmitter has an individual power constraint and can obtain their own perfect channel state information (CSI). The transmitters first cooperate by sharing the CSI, and then jointly optimize power allocation in the metric of sum throughput, which can be modeled as a non-convex constrained optimization problem. Through an application of Karush-Kuhn-Tucker conditions, the problem is reformulated as a convex one. Then, the closed form solution is derived with the nature of traditional WF as well as cooperative properties. Based on the derived solution, an iteration algorithm for joint water level is given for the first time, which can be explained as a cooperative WF relative to the traditional WF. Motivated by the deriving process, we extend parts of the conclusion to N-transmitter case. Numerical results are presented to evaluate the optimal power allocation scheme in OFDM cellular system. For comparison, we also evaluate the traditional non-cooperative WF and equal power allocation scheme.

Evolutionary auto-tuning algorithm for PID controllers

In this work, an auto-tuning procedure for PID controllers is presented. This autotuning procedure identifies a FOPDT model for a given process and uses an evolutionary algorithm to solve a constrained non-convex optimization problem to adjust the parameters of a PID controller with derivative filter. The auto-tuning procedure is validated with a set of process with different characteristics. Presented results validate the auto-tuning algorithm as a practical and viable application for auto-tuning procedures.