Difference Of Convex Programming Research Articles

Probability rate optimization of positive Markov jump linear systems via DC programming

AbstractWe investigate the stabilization problem of positive Markov jump linear systems by optimizing their transition probability rates. By using the convex property of posynomials and the standard mathematical programming that deals with the difference in convex functions, we show that transition probability rate synthesis problems can be solved via difference‐of‐convex (DC) programming. A numerical example is used to illustrate the effectiveness of our results.

The Descent–Ascent Algorithm for DC Programming

We introduce a bundle method for the unconstrained minimization of nonsmooth difference-of-convex (DC) functions, and it is based on the calculation of a special type of descent direction called descent–ascent direction. The algorithm only requires evaluations of the minuend component function at each iterate, and it can be considered as a parsimonious bundle method as accumulation of information takes place only in case the descent–ascent direction does not provide a sufficient decrease. No line search is performed, and proximity control is pursued independent of whether the decrease in the objective function is achieved. Termination of the algorithm at a point satisfying a weak criticality condition is proved, and numerical results on a set of benchmark DC problems are reported. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms – Continuous. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0142 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0142 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

An Accelerated Double-Proximal Gradient Algorithm for DC Programming

The double-proximal gradient algorithm (DPGA) is a new variant of the classical difference-of-convex algorithm (DCA) for solving difference-of-convex (DC) optimization problems. In this paper, we propose an accelerated version of the double-proximal gradient algorithm for DC programming, in which the objective function consists of three convex modules (only one module is smooth). We establish convergence of the sequence generated by our algorithm if the objective function satisfies the Kurdyka–[Formula: see text]ojasiewicz (K[Formula: see text]) property and show that its convergence rate is not weaker than DPGA. Compared with DPGA, the numerical experiments on an image processing model show that the number of iterations of ADPGA is reduced by 43.57% and the running time is reduced by 43.47% on average.

A bundle-type method for nonsmooth DC programs

A bundle method for minimizing the difference of convex (DC) and possibly nonsmooth functions is developed. The method may be viewed as an inexact version of the DC algorithm, where each subproblem is solved only approximately by a bundle method. We always terminate the bundle method after the first serious step. This yields a descent direction for the original objective function, and it is shown that a stepsize of at least one is accepted in this way. Using a line search, even larger stepsizes are possible. The overall method is shown to be globally convergent to critical points of DC programs. The new algorithm is tested and compared to some other solution methods on several examples and realistic applications.

Abort orbit determination of launch vehicles based on difference of convex programming

Existing exoatmospheric guidance laws become ineffective when a thrust loss fault (like one or more engines unexpectedly shut down) occurs in a launch vehicle, which makes the nominal orbit unreachable. This paper presents a new abort orbit determination process to determine an abort orbit which the launch vehicle is able to reach. It involves solving at most two optimization problems and the abort orbit may be circular or elliptical. We will present how to transform the nonconvex optimization problems into difference of convex (DC) programming problems which can be solved by existing methods with guaranteed convergence. The main contribution of this paper lies in proposing a relaxation technique to convexify a nonconvex control constraint and the combination of the introduction of new constraints and a relaxation technique to deal with the strict terminal constraints. Validity of the relaxation technique is ensured by theoretical analysis. A unique feature of the transformation process is that the feasible sets defined by both the control constraint and the strict terminal constraints are all enlarged. This brings a critical benefit that the resulting DC programming problems can be very efficiently solved. Thus, an abort orbit determination algorithm, which is based on DC programming, can be designed to very reliably and efficiently determine an abort orbit, which will be demonstrated by numerical examples.

Deep Reinforcement Learning-Aided Optimization of Multi-Interface Allocation for Short-Packet Communications

The severe spectrum scarcity and the stringent requirements of Beyond 5G applications call for an integrated use of low frequency Sub-6 GHz and high frequency millimeter Wave bands. Focusing on future Internet of Things (IoT) Short-Packet Communications (SPC), this paper investigates the optimized usage of such diverse wireless interfaces. We propose an unifying framework devoted to SPC that jointly optimizes the user partitioning over each band, and the radio resource scheduling within each band. Leveraging Deep Reinforcement Learning (DRL) tools, the proposed method enables to better tackle the challenges imposed by dynamically varying mobile environments such as the Line-of-Sight situations of each link, and the heterogeneity of individual Quality of Service (QoS) requirements, such as rate, delay and reliability. Regarding the DRL-based user partitioning to each band, we have investigated three different types of partitioning actions to obtain a high network performance as well as a rapid convergence. Regarding the proposed sub-schedulers within each band, we designed two optimization methods, i.e., one that leverages Difference of Convex Programming (DCP) technique, and the second that accelerates convergence to a local optimum. Numerical evaluations show that the proposed methods outperform conventional approaches in terms of sum-rate and QoS outage probabilities.

On the Mathematical Modeling and Optimization for the Energy Efficiency Performance of CSMA-NOMA Random Access Networks With Channel Inversion

The paradigm of upcoming 5G and beyond is the massive machine type communications (mMTC), where a large number of devices automatically operate wireless communications. Due to their automatic and energy-intensive operations, energy efficiency (EE) becomes crucial, but there is a lack of investigation for EE of random access networks, which is the underlying platform for mMTC. In this paper, we focus on the EE of carrier sense multiple access-based non-orthogonal multiple access (NOMA) random access networks. We first construct mathematical models. Instead of investigating all combinations regarding successful decoding events as in previous works, by pivoting around the decoding process of a specific signal, a hidden pattern of NOMA decoding process is unveiled, which can largely decrease analytical complexity. Then, together with this feature, by adopting Markov chain and Q-function approximation, closed-form formulation for EE is derived. Subsequently, to efficiently solve the complicated non-convex EE maximization problem built via the constructed models, we employ an approach that unifies complementary geometric programming (CGP) and difference of convex programming (DCP) to optimize all the controllable parameters at device side, namely, transmission probability, power, and data rate, with a tightest lower bound strategy to guarantee seamless EE improvement and very fast convergence to local optimal points even in the worst case. Simulation experiments verify the accuracy of mathematical models and efficiency of optimization scheme.

A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling

This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as a nonlocal reparametrization of the assignment flow approach that was introduced in [J. Math. Imaging Vision, 58 (2017), pp. 211–238]. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference of convex (DC) functions decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments.

Retraction-based first-order feasible methods for difference-of-convex programs with smooth inequality and simple geometric constraints

In this paper, we propose first-order feasible methods for difference-of-convex (DC) programs with smooth inequality and simple geometric constraints. Our strategy for maintaining feasibility of the iterates is based on a “retraction” idea adapted from the literature of manifold optimization. When the constraints are convex, we establish the global subsequential convergence of the sequence generated by our algorithm under strict feasibility condition, and analyze its convergence rate when the objective is in addition convex according to the Kurdyka-Łojasiewicz (KL) exponent of the extended objective (i.e., sum of the objective and the indicator function of the constraint set). We also show that the extended objective of a large class of Euclidean norm (and more generally, group LASSO penalty) regularized convex optimization problems is a KL function with exponent $\frac {1}{2}$ ; consequently, our algorithm is locally linearly convergent when applied to these problems. We then extend our method to solve DC programs with a single specially structured nonconvex constraint. Finally, we discuss how our algorithms can be applied to solve two concrete optimization problems, namely, group-structured compressed sensing problems with Gaussian measurement noise and compressed sensing problems with Cauchy measurement noise, and illustrate the empirical performance of our algorithms.

A variable metric and nesterov extrapolated proximal DCA with backtracking for a composite DC program

In this paper, we consider a composite difference-of-convex (DC) program, whose objective function is the sum of a smooth convex function with Lipschitz continuous gradient, a proper closed and convex function, and a continuous concave function. This problem has many applications in machine learning and data science. The proximal DCA (pDCA), a special case of the classical difference-of-convex algorithm (DCA), as well as two Nesterov-type extrapolated DCA – ADCA (Phan et al. IJCAI:1369–1375, 2018) and pDCAe (Wen et al. Comput. Optim. Appl. 69:297–324, 2018) – can solve this problem. The algorithmic stepsizes of pDCA, pDCAe, and ADCA are fixed and determined by estimating a prior the smoothness parameter of the loss function. However, such an estimate may be hard to obtain or poor in some real-world applications. Motivated by this difficulty, we propose a variable metric and Nesterov extrapolated proximal DCA with backtracking (SPDCAe), which combines the backtracking line search procedure (not necessarily monotone) and the Nesterov's extrapolation for potential acceleration; moreover, the variable metric method is incorporated for better local approximation. Numerical simulations on sparse binary logistic regression and compressed sensing with Poisson noise demonstrate the effectiveness of our proposed method.

Algorithms for Difference-of-Convex Programs Based on Difference-of-Moreau-Envelopes Smoothing

In this paper, we consider minimization of a difference-of-convex (DC) function with and without linear equality constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, capture stationary points, local, and global minima of the original DC function, and enjoy some growth conditions, such as level-boundedness and coercivity, for broad classes of DC functions. For a smoothed DC program without linear constraints, it is shown that the classic gradient descent method and an inexact variant converge to a stationary solution of the original DC function in the limit with a rate of [Formula: see text], where K is the number of proximal evaluations of both components. Furthermore, when the DC program is explicitly constrained in an affine subspace, we combine the smoothing technique with the augmented Lagrangian function and derive two variants of the augmented Lagrangian method (ALM), named linearly constrained DC (LCDC)-ALM and composite LCDC-ALM, targeting on different structures of the DC objective function. We show that both algorithms find an ϵ-approximate stationary solution of the original DC program in [Formula: see text] iterations. Comparing to existing methods designed for linearly constrained weakly convex minimization, the proposed ALM-based algorithms can be applied to a broader class of problems, where the objective contains a nonsmooth concave component. Finally, numerical experiments are presented to demonstrate the performance of the proposed algorithms. Funding: This work was partially supported by the NSF [Grant ECCS1751747]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoo.2022.0087 .

Three-Dimensional Cooperative Positioning for Internet of Things Provenance

A large number of Internet of Things (IoT) devices have been interconnected for information collection and exchange. The data are only meaningful if it is captured at the expected location (i.e., the IoT devices or sensors are not removed accidentally or intentionally). This article presents a new algorithm, which cooperatively locates multiple IoT devices deployed in a 3-D space based on pairwise Euclidean distance measurements. When the distance measurement noises are negligible, a new feasibility problem of rank-3 variables is formulated. We solve the problem using the difference-of-convex (DC) programming to preserve the rank-3 constraints, rather than relaxing the constraints, using semidefinite relaxation (SDR). When the distance measurements are corrupted by additive noises and nonlight-of-sight (NLOS) propagation, a maximum-likelihood estimation (MLE) problem is formulated and transformed to a DC program solved with the rank-3 constraints preserved. Simulation results indicate that the proposed approach can achieve satisfactory accuracy results with a low complexity and strong robustness to the irregular topology, poor connectivity, and measurement errors, as compared to existing SDR-based alternatives.

Confidence bands for a log-concave density

We present a new approach for inference about a univariate log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf F. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for F which we introduce here has attractive computational and statistical properties: It allows to bring modern tools from optimization to bear on this problem via difference of convex programming, and it results in optimal statistical inference. We show that the width of the resulting confidence bands converges at nearly the parametric rate when the log density is k-affine.

An Inexact Proximal DC Algorithm with Sieving Strategy for Rank Constrained Least Squares Semidefinite Programming

In this paper, the optimization problem of supervised distance preserving projection (SDPP) for data dimensionality reduction is considered, which is equivalent to a rank constrained least squares semidefinite programming (RCLSSDP). Due to the combinatorial nature of rank function, the rank constrained optimization problems are NP-hard in most cases. In order to overcome the difficulties caused by rank constraint, a difference-of-convex (DC) regularization strategy is employed, then RCLSSDP is transferred into a DC programming. For solving the corresponding DC problem, an inexact proximal DC algorithm with sieving strategy (s-iPDCA) is proposed, whose subproblems are solved by an accelerated block coordinate descent method. The global convergence of the sequence generated by s-iPDCA is proved. To illustrate the efficiency of the proposed algorithm for solving RCLSSDP, s-iPDCA is compared with classical proximal DC algorithm, proximal gradient method, proximal gradient-DC algorithm and proximal DC algorithm with extrapolation by performing dimensionality reduction experiment on COIL-20 database. From the computation time and the quality of solution, the numerical results demonstrate that s-iPDCA outperforms other methods. Moreover, dimensionality reduction experiments for face recognition on ORL and YaleB databases demonstrate that rank constrained kernel SDPP is efficient and competitive when comparing with kernel semidefinite SDPP and kernel principal component analysis in terms of recognition accuracy.

Distributed chance-constrained based total energy supply capability evaluation method for integrated power and natural gas system

With the development of gas turbines and power to gas technologies, the modern energy infrastructure has evolved into a coupled power and natural gas system. To fully exploit the maximum total energy supply capability (TESC) of the integrated power distribution and natural gas system (IDGS), a distributed chance-constrained based TESC evaluation method with consideration of wind power uncertainties is proposed in this paper, which can be utilized as a guideline for the planning, operation and reliability assessment of the IDGS. Specifically, to preserve the privacy of the operation data shared among power distribution system (PDS) and natural gas system (NGS), the established TESC model is solved in a distributed manner based on the analytical target cascading (ATC) algorithm. In the PDS and NGS sub-problems, the difference-of-convex programming (DCP) algorithm is utilized to deal with the nonconvex power flow and Weymouth gas flow equations to guarantee the feasibility of the solution results. Moreover, the uncertain nature of wind power is incorporated into the model and represented by a set of joint chance constraints (JCCs) with a predefined confidence level, which can be effectively solved via a novel Boolean reformulation method. Numerical studies are performed on a modified 94-bus distribution system and 20-node natural gas system to verify the effectiveness of the proposed model and solution methodology.

A refined inertial DC algorithm for DC programming

In this paper we consider the difference-of-convex (DC) programming problems, whose objective function is the difference of two convex functions. The classical DC Algorithm (DCA) is well-known for solving this kind of problems, which generally returns a critical point. Recently, an inertial DC algorithm (InDCA) equipped with heavy-ball inertial-force procedure was proposed in de Oliveira et al. (Set-Valued and Variational Analysis 27(4):895--919, 2019), which potentially helps to improve both the convergence speed and the solution quality. Based on InDCA, we propose a refined inertial DC algorithm (RInDCA) equipped with enlarged inertial step-size compared with InDCA. Empirically, larger step-size accelerates the convergence. We demonstrate the subsequential convergence of our refined version to a critical point. In addition, by assuming the Kurdyka-{\L}ojasiewicz (KL) property of the objective function, we establish the sequential convergence of RInDCA. Numerical simulations on checking copositivity of matrices and image denoising problem show the benefit of larger step-size.

Penalty and Augmented Lagrangian Methods for Constrained DC Programming

In this paper, we consider a class of structured nonsmooth difference-of-convex (DC) constrained DC programs in which the first convex component of the objective and constraints is the sum of a smooth and a nonsmooth function, and their second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have a weak convergence guarantee or require a feasible initial point. Inspired by the recent work by Pang et al. [Pang J-S, Razaviyayn M, Alvarado A (2017) Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. 42(1):95–118.], in this paper, we propose two infeasible methods with a strong convergence guarantee for the considered problem. The first one is a penalty method that consists of finding an approximate D-stationary point of a sequence of penalty subproblems. We show that any feasible accumulation point of the solution sequence generated by such a penalty method is a B-stationary point of the problem under a weakest possible assumption that it satisfies a pointwise Slater constraint qualification (PSCQ). The second one is an augmented Lagrangian (AL) method that consists of finding an approximate D-stationary point of a sequence of AL subproblems. Under the same PSCQ condition as for the penalty method, we show that any feasible accumulation point of the solution sequence generated by such an AL method is a B-stationary point of the problem, and moreover, it satisfies a Karush–Kuhn–Tucker type of optimality condition for the problem, together with any accumulation point of the sequence of a set of auxiliary Lagrangian multipliers. We also propose an efficient successive convex approximation method for computing an approximate D-stationary point of the penalty and AL subproblems. Finally, some numerical experiments are conducted to demonstrate the efficiency of our proposed methods.

Decentralized closed-loop optimization for 6-DOF self-assembly satellites

A decentralized closed-loop optimization based on the difference of convex programming is proposed to assemble large telescopes autonomously in-orbit. This study focuses on assembly problems of mirror satellites with six degrees of freedom. Firstly, the hp-Radau pseudospectral collocation method is adopted to increase computational efficiency. Subsequently, the nonconvex constraints are convexified by applying undominated decomposition and sequential convex programming. Additionally, the fixed trajectory assumption enables the translational optimization problem to be a decentralized problem with convergence-guaranteed proof. The six degrees of freedom optimization problem is decoupled by utilizing the proposed algorithm in translational and rotational planning, and coupled by thruster allocation method. To improve the robustness of the system, the algorithm is wrapped in the model predictive control framework, and the recursive feasible property is ensured using the outer-bounding tube. Therefore, the closed-loop optimal control is robust under additive uncertainties. In the numerical experiments, mirror satellites are released from a high-accuracy fitting carrier satellite to accomplish the assembly, and these satellites are utilized to verify the high-speed algorithm and the effectiveness of the closed-loop optimization.

Joint Beamforming and Power-Splitting Design for Cooperative Nonorthogonal Multicast

We propose a cooperative nonorthogonal multicast scheme for multiple-input-single-output (MISO) systems, where the transmitter sends the superimposed signal to two multicast groups. After successfully detecting all signals, the cell-center users (CCUs) in one multicast group help relay signal to the cell-edge users (CEUs) in another group to enhance signal reception. Simultaneous wireless information and power transfer is adopted at CCUs to assist information relaying. In such a system, to minimize the transmission power of the system while satisfying the quality of service requirements of all the users, an optimization problem is formulated to jointly design the transmitter beamformers, distributed CCUs beamformer, as well as the power-splitting (PS) ratios. To solve the nonconvex problem, we first equivalently transform it into a difference of convex (DC) programming. Then, a low-complexity iterative algorithm based on the constrained concave convex procedure (CCCP) is proposed to solve the DC programming one. In addition, a robust joint beamforming and PS scheme is proposed by assuming imperfect channel state information (CSI). The infinite constraints caused by CSI uncertainties are converted into finite ones utilizing the S-procedure. To obtain a rank-one locally optimal solution, the penalty function method and CCCP algorithm are adopted. The simulation results reveal that the proposed cooperative scheme can greatly reduce the transmission power and outperform the baselines within a certain range. Moreover, the robustness and effectiveness of the robust design under imperfect CSI case have been validated.

A novel approach for solving stochastic problems with multiple objective functions

In this paper we suggest an approach for solving a multiobjective stochastic linear programming problem with normal multivariate distributions. Our approach is a combination between a multiobjective method and a nonconvex technique. The problem is first transformed into a deterministic multiobjective problem introducing the expected value criterion and an utility function that represents the decision makers preferences. The obtained problem is reduced to a mono-objective quadratic problem using a weighting method. This last problem is solved by DC (Difference of Convex) programming and DC algorithm. A numerical example is included for illustration.