Solving Convex Optimization Research Articles

Discrete Convex Optimization Solvers and Demonstration Softwares

Discrete Convex Optimization Solvers and Demonstration Softwares

Quadratic approximate dynamic programming for input‐affine systems

SUMMARYWe consider the use of quadratic approximate value functions for stochastic control problems with input‐affine dynamics and convex stage cost and constraints. Evaluating the approximate dynamic programming policy in such cases requires the solution of an explicit convex optimization problem, such as a quadratic program, which can be carried out efficiently. We describe a simple and general method for approximate value iteration that also relies on our ability to solve convex optimization problems, in this case, typically a semidefinite program. Although we have no theoretical guarantee on the performance attained using our method, we observe that very good performance can be obtained in practice.Copyright © 2012 John Wiley & Sons, Ltd.

Optimal Error Correction and Methods of Feasible Directions

The main objective of this study is to discuss the optimum correction of linear inequality systems and absolute value equations (AVE). In this work, a simple and efficient feasible direction method will be provided for solving two fractional nonconvex minimization problems that result from the optimal correction of a linear system. We will show that, in some special-but frequently encountered-cases, we can solve convex optimization problems instead of not-necessarily-convex fractional problems. And, by using the method of feasible directions, we solve the optimal correction problem. Some examples are provided to illustrate the efficiency and validity of the proposed method.

Design of Pre-Rake DS-UWB Downlink with Pre-Equalization

We consider the design of ultra-wideband (UWB) systems that enable high data rate communications for short-range wireless applications. In particular, we consider the downlink of a direct sequence UWB (DS-UWB) system in which the base station is equipped with multiple antennas and employs pre-rake combining, while each user employs a simple single antenna receiver. We propose the use of multiuser filters for the purpose of pre-equalization at the transmitter in order to mitigate the combined effects of intersymbol interference (ISI) and multiuser interference (MUI) that are generated at the receivers as a result of the wideband nature of the users' channels. For this system, we study the joint design of the transmitter's pre-equalization filters and each receiver's scalar gain under two design criteria. The first design minimizes the total transmitted power from the base station subject to achieving physical layer quality of service requirements of different users. For this design, we show that the calculation of the pre-equalization filters and the receiver gains can be formulated as an efficiently solvable convex optimization problem. In the second design, we consider the minimization of a weighted sum of each user's mean-square error. In order to obtain a computationally tractable solution for this design criterion, we exploit the dual DS-UWB uplink that employs rake combining and post-equalization filters at a central receiver. The numerical studies for each design criterion under realistic models of UWB channel propagation demonstrate the effectiveness of the proposed multiuser pre-equalization filter designs in mitigating ISI and MUI, and thus their ability to enable reliable pre-rake DS-UWB downlink transmission.

Sparse Signal Recovery via ECME Thresholding Pursuits

The emerging theory of compressive sensing (CS) provides a new sparse signal processing paradigm for reconstructing sparse signals from the undersampled linear measurements. Recently, numerous algorithms have been developed to solve convex optimization problems for CS sparse signal recovery. However, in some certain circumstances, greedy algorithms exhibit superior performance than convex methods. This paper is a followup to the recent paper of Wang and Yin (2010), who refine BP reconstructions via iterative support detection (ISD). The heuristic idea of ISD was applied to greedy algorithms. We developed two approaches for accelerating the ECME iteration. The described algorithms, named ECME thresholding pursuits (EMTP), introduced two greedy strategies that each iteration detects a support set I by thresholding the result of the ECME iteration and estimates the reconstructed signal by solving a truncated least‐squares problem on the support set I. Two effective support detection strategies are devised for the sparse signals with components having a fast decaying distribution of nonzero components. The experimental studies are presented to demonstrate that EMTP offers an appealing alternative to state‐of‐the‐art algorithms for sparse signal recovery.

Predictive Control for Polynomial Systems Subject to State and Input Constraints

Abstract We present a nonlinear model predictive control (NMPC) approach to design optimal feedback controllers for the class of continuous-time input affine polynomial systems subject to state and input constraints. The corresponding feedback law is obtained via the solution of an efficient to solve convex optimization problem subject to sum of squares (SOS) constraints. To serve the conflicting requirements of performance and computational costs in various applications, the approach comes in three different control schemes which differ in how many times the (updated) optimization problem has to be solved, and whether this is done on- or offline. Each of the proposed control schemes minimizes an upper bound on the cost functional while guaranteeing closed-loop stability and satisfaction of input and state constraints. Finally, simulations of an example system show the applicability and the effectiveness of the proposed controllers.

Image decoding optimization based on compressive sensing

Transform-based image codec follows the basic principle: the reconstructed quality is decided by the quantization level. Compressive sensing (CS) breaks the limit and states that sparse signals can be perfectly recovered from incomplete or even corrupted information by solving convex optimization. Under the same acquisition of images, if images are represented sparsely enough, they can be reconstructed more accurately by CS recovery than inverse transform. So, in this paper, we utilize a modified TV operator to enhance image sparse representation and reconstruction accuracy, and we acquire image information from transform coefficients corrupted by quantization noise. We can reconstruct the images by CS recovery instead of inverse transform. A CS-based JPEG decoding scheme is obtained and experimental results demonstrate that the proposed methods significantly improve the PSNR and visual quality of reconstructed images compared with original JPEG decoder.

LMI Techniques for Optimization Over Polynomials in Control: A Survey

Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Polya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.

Guaranteed performance state estimation of static neural networks with time-varying delay

This paper is concerned with studying two kinds of guaranteed performance state estimation problems for static neural networks with time-varying delay. Both delay-independent and delay-dependent design criteria are presented under which the resulting estimation error system is globally asymptotically stable and a prescribed performance is guaranteed in the H ∞ or generalized H 2 sense. It is shown that the gain matrices of the state estimator and the optimal performance indexes can be simultaneously obtained by solving convex optimization problems subject to linear matrix inequalities. It is worth noting that no slack variable is introduced in the proposed conditions, and thus the computational burden is reduced. The effectiveness of the developed results is finally demonstrated by simulation examples.

A partial parallel splitting augmented Lagrangian method for solving constrained matrix optimization problems

Alternating directions methods (ADMs) are very effective for solving convex optimization problems with separable structure. However, when these methods are applied to solve convex optimization problems with three separable operators, their convergence results have not been established as yet. In this paper, we consider a class of constrained matrix optimization problems. The problem is first reformulated into a convex optimization problem with three separable operators, then it is solved by a proposed partial parallel splitting method. The proposed method combines the parallel splitting (augmented Lagrangian) method (PSALM) and the alternating directions method (ADM), and it is referred to as PADALM in short. The main difference between PADALM and PSALM is that in PADALM, two operators are handled first by a parallel method, then the third operator and the former two are dealt with by an alternating method. Finally, the convergence result for PADALM is established and numerical results are provided to show the efficacy of PADALM and its superiority over PSALM.

Efficient design of rotary traveling wave oscillator array via geometric programming

This paper presents an efficient method for globally optimizing and automating component sizing for rotary traveling wave oscillator arrays. The lumped equivalent model of transmission lines loaded by inverter pairs is evaluated and posynomial functions for oscillation frequency, power dissipation, phase noise, etc. are formulated using transmission line theory. The resulting design problem can be posed as a geometric programming problem, which can be efficiently solved with a convex optimization solver. The proposed method can compute the global optima more efficiently than the traditional iterative scheme and various design problems can be solved with the same circuit model. The globally optimal trade-off curves between competing objectives are also computed to carry out robust designs and quickly explore the design space.

The scenario approach for systems and control design

The ‘scenario approach’ is an innovative technology that has been introduced to solve convex optimization problems with an infinite number of constraints, a class of problems which often occurs when dealing with uncertainty. This technology relies on random sampling of constraints, and provides a powerful means for solving a variety of design problems in systems and control. The objective of this paper is to illustrate the scenario approach at a tutorial level, focusing mainly on algorithmic aspects. Its versatility and virtues will be pointed out through a number of examples in model reduction, robust and optimal control.

The minimum heat consumption for heat-driven binary separation processes with linear phenomenological heat transfer law

The optimal performance of heat-driven binary separation processes with linear phenomenological heat transfer law (q∝Δ(T −1)) is analyzed by taking the processes as heat engines which work between high- and low-temperature reservoirs and produce enthalpy and energy flows out of the system, and the temperatures of the heat reservoirs are assumed to be time- and space-variables. A numerical method is employed to solve convex optimization problem and Lagrangian function is employed to solve the average optimal control problem. The dimensionless entropy production rate coefficient and dimensionless enthalpy flow rate coefficient are adopted to indicate the major influence factors on the performance of the separation process, such as the properties of different materials and various separation requirements for the separation process. The dimensionless minimum average entropy production rate and dimensionless minimum average heat consumption of the heat-driven binary separation processes are obtained. The obtained results are compared with those obtained with the Newtonian heat transfer law (q∝Δ(T)).

A cutting plane method for solving convex optimization problems over the cone of nonnegative polynomials

Many practical problems can be formulated as convex optimization problems over the cone of nonnegative univariate polynomials. We use a cutting plane method for solving this type of optimization pr...

Control Liapunov function design of neural networks that solve convex optimization and variational inequality problems

This paper presents two neural networks to find the optimal point in convex optimization problems and variational inequality problems, respectively. The domain of the functions that define the problems is a convex set, which is determined by convex inequality constraints and affine equality constraints. The neural networks are based on gradient descent and exact penalization and the convergence analysis is based on a control Liapunov function analysis, since the dynamical system corresponding to each neural network may be viewed as a so-called variable structure closed loop control system.

Optimal Distributed Stochastic Routing Algorithms for Wireless Multihop Networks

A novel framework was introduced recently for stochastic routing in wireless multihop networks, whereby each node selects a neighbor to forward a packet according to a probability distribution. Generalizing (deterministic) shortest path routing, stochastic routing offers greater flexibility that matches the random nature of wireless links. Consider the pairwise reliability matrix R, whose (i, j)-th entry Rij represents the probability that a packet transmitted from the j-th user Uj is correctly received by the i-th user Ui. Using R to capture physical layer aspects of the wireless medium, several rate-oriented stochastic routing formulations can be reduced to centrally solvable convex optimization problems. The present paper, introduces distributed algorithms that find optimal routing probabilities without the burden of collecting R at a central node and then percolating the resulting routing probabilities through network nodes. The resultant schemes are distributed in the sense that: (i) terminal Uj has access only to the j-th row and column of R; and (ii) Uj interchanges variables only with those single-hop neighbors having positive probability of decoding its packets. The distributed algorithms are built by recasting the optimization problems and applying dual decomposition techniques. Since iterates obtained via dual decomposition do not always converge to centralized optimal routing probabilities, two known regularization approaches are further invoked, namely the method of multipliers (MoM) and the alternating-direction MoM. Convergence to the optimal routing matrix is then guaranteed under mild conditions. Many rate-oriented optimality criteria of practical interest can be addressed by the distributed framework, including maximization of: (i) the minimum rate; (ii) a weighted sum of rates; (iii) the product of rates; and (iv) the source's rate in a relay network. Robustness of the distributed algorithms is tested with respect to topological changes, communication errors and node mobility.

Computing sum of squares decompositions with rational coefficients

Sum of squares (SOS) decompositions for nonnegative polynomials are usually computed numerically, using convex optimization solvers. Although the underlying floating point methods in principle allow for numerical approximations of arbitrary precision, the computed solutions will never be exact. In many applications such as geometric theorem proving, it is of interest to obtain solutions that can be exactly verified. In this paper, we present a numeric–symbolic method that exploits the efficiency of numerical techniques to obtain an approximate solution, which is then used as a starting point for the computation of an exact rational result. We show that under a strict feasibility assumption, an approximate solution of the semidefinite program is sufficient to obtain a rational decomposition, and quantify the relation between the numerical error versus the rounding tolerance needed. Furthermore, we present an implementation of our method for the computer algebra system Macaulay 2.

Gradient Pursuits

Sparse signal approximations have become a fundamental tool in signal processing with wide-ranging applications from source separation to signal acquisition. The ever-growing number of possible applications and, in particular, the ever-increasing problem sizes now addressed lead to new challenges in terms of computational strategies and the development of fast and efficient algorithms has become paramount. Recently, very fast algorithms have been developed to solve convex optimization problems that are often used to approximate the sparse approximation problem; however, it has also been shown, that in certain circumstances, greedy strategies, such as orthogonal matching pursuit, can have better performance than the convex methods. In this paper, improvements to greedy strategies are proposed and algorithms are developed that approximate orthogonal matching pursuit with computational requirements more akin to matching pursuit. Three different directional optimization schemes based on the gradient, the conjugate gradient, and an approximation to the conjugate gradient are discussed, respectively. It is shown that the conjugate gradient update leads to a novel implementation of orthogonal matching pursuit, while the gradient-based approach as well as the approximate conjugate gradient methods both lead to fast approximations to orthogonal matching pursuit, with the approximate conjugate gradient method being superior to the gradient method.

Multiple Lyapunov Function Based Reaching Condition for Orbital Existence of Switching Power Converters

A methodology to analyze the reaching condition of a switching power converter (SPC) using Lyapunov's direct method and a piecewise linear model is outlined. By using a multiple Lyapunov function, the reaching criteria for orbital existence of a SPC is formulated as a linear matrix inequality that is solved using a convex optimization solver. Further, the criterion is modified to distinguish the different modes (i.e., sliding and asymptotic modes and combination of the two fundamental modes) of convergences of the reaching dynamics. The applicability of the reaching criteria for solving practical problems using case illustrations of dc-dc converters and three-phase dc-ac converters operating with different control and modulation techniques is demonstrated. The methodology developed in this paper can be potentially extended to other SPCs and may lead to the development of optimal-sequence control techniques that can dynamically change the mode and the rate of convergence of a SPC.

The Scenario Approach for Systems and Control Design

The ‘scenario approach’ is an innovative technology that has been introduced to solve convex optimization problems with an infinite number of constraints, a class of problems which often occurs when dealing with uncertainty. This technology relies on random sampling of constraints, and provides a powerful means for solving a variety of design problems in systems and control. The objective of this paper is to illustrate the scenario approach at a tutorial level, focusing mainly on algorithmic aspects. Specifically, its versatility and virtues will be pointed out through a number of examples in model reduction, robust and optimal control.