Nonconvex Terms Research Articles

Evolutionary variational\u2013hemivariational inequalities with applications to dynamic viscoelastic contact mechanics

The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.

Leader-Following Consensus for Multi-Agent Systems With Asynchronous Control Modes Under Nonhomogeneous Markovian Jump Network Topology

Under a nonhomogeneous Markovian jump network topology, this paper addresses the problem of leader-following consensus for multi-agent systems with an asynchronous control mode. Specifically, based on an asynchronous mode-dependent Lyapunov function and a suitable relaxation process, sufficient conditions for leader-following consensus are derived in such a way that 1) non-convex terms multiplied by time-varying signed parameters can be addressed, 2) the emergence of an asynchronous control mode can be reflected in the control synthesis of the follower agents, and 3) the constraints of two combined parameters can be explicitly imposed on the process of transforming parameterized linear matrix inequalities (PLMIs) into LMIs. Finally, two illustrative examples are given to show the validity of the proposed method.

A convex-nonconvex variational method for the additive decomposition of functions on surfaces

We present a convex-nonconvex variational approach for the additive decomposition of noisy scalar fields defined over triangulated surfaces into piecewise constant and smooth components. The energy functional to be minimized is defined by the weighted sum of three terms, namely an fidelity term for the noise component, a Tikhonov regularization term for the smooth component and a total variation (TV)-like non-convex term for the piecewise constant component. The last term is parametrized such that the free scalar parameter allows to tune its degree of non-convexity and, hence, to separate the piecewise constant component more effectively than by using a classical convex TV regularizer without renouncing to convexity of the total energy functional. A method is also presented for selecting the two regularization parameters. The unique solution of the proposed variational model is determined by means of an efficient ADMM-based minimization algorithm. Numerical experiments show a nearly perfect separation of the different components.

ADMM for multiaffine constrained optimization

ABSTRACTWe expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka–Łojasiewicz (K–Ł) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavoured to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnegative matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions.

Laplacian-Uniform Mixture-Driven Iterative Robust Coding With Applications to Face Recognition Against Dense Errors.

Outliers due to occlusion, pixel corruption, and so on pose serious challenges to face recognition despite the recent progress brought by sparse representation. In this article, we show that robust statistics implemented by the state-of-the-art methods are insufficient for robustness against dense gross errors. By modeling the distribution of coding residuals with a Laplacian-uniform mixture, we obtain a sparse representation that is significantly more robust than the previous methods. The nonconvex error term of the implemented objective function is nondifferentiable at zero and cannot be properly addressed by the usual iteratively reweighted least-squares formulation. We show that an iterative robust coding algorithm can be derived by local linear approximation of the nonconvex error term, which is both effective and efficient. With iteratively reweighted l1 minimization of the error term, the proposed algorithm is capable of handling the sparsity assumption of the coding errors more appropriately than the previous methods. Notably, it has the distinct property of addressing error detection and error correction cooperatively in the robust coding process. The proposed method demonstrates significantly improved robustness for face recognition against dense gross errors, either contiguous or discontiguous, as verified by extensive experiments.

Sparsity promoting regularization for effective noise suppression in SPECT image reconstruction

The purpose of this research is to develop an advanced reconstruction method for low-count, hence high-noise, single-photon emission computed tomography (SPECT) image reconstruction. It consists of a novel reconstruction model to suppress noise while conducting reconstruction and an efficient algorithm to solve the model. A novel regularizer is introduced as the nonconvex denoising term based on the approximate sparsity of the image under a geometric tight frame transform domain. The deblurring term is based on the negative log-likelihood of the SPECT data model. To solve the resulting nonconvex optimization problem a preconditioned fixed-point proximity algorithm (PFPA) is introduced. We prove that under appropriate assumptions, PFPA converges to a local solution of the optimization problem at a global convergence rate. Substantial numerical results for simulation data are presented to demonstrate the superiority of the proposed method in denoising, suppressing artifacts and reconstruction accuracy. We simulate noisy 2D SPECT data from two phantoms: hot Gaussian spheres on random lumpy warm background, and the anthropomorphic brain phantom, at high- and low-noise levels (64k and 90k counts, respectively), and reconstruct them using PFPA. We also perform limited comparative studies with selected competing state-of-the-art total variation (TV) and higher-order TV (HOTV) transform-based methods, and widely used post-filtered maximum-likelihood expectation-maximization. We investigate imaging performance of these methods using: contrast-to-noise ratio (CNR), ensemble variance images (EVI), background ensemble noise (BEN), normalized mean-square error (NMSE), and channelized hotelling observer (CHO) detectability. Each of the competing methods is independently optimized for each metric. We establish that the proposed method outperforms the other approaches in all image quality metrics except NMSE where it is matched by HOTV. The superiority of the proposed method is especially evident in the CHO detectability tests results. We also perform qualitative image evaluation for presence and severity of image artifacts where it also performs better in terms of suppressing ‘staircase’ artifacts, as compared to TV methods. However, edge artifacts on high-contrast regions persist. We conclude that the proposed method may offer a powerful tool for detection tasks in high-noise SPECT imaging.

Indicator of power convex and exponential transformations for solving nonlinear problems containing posynomial terms

Posynomial terms frequently appear in many nonlinear problems and are the core components of geometric and generalized geometric programming problems. The most popular method to treat nonconvex posynomial terms for obtaining global optimization is to convert nonconvex posynomial terms as convex underestimators using transformation techniques. Among the transformation techniques, exponential transformation (ET) and power convex transformation (PCT) can yield the tightest underestimators of posynomial terms. However, the current literature has rarely discussed which to select between ET and PCT. This study employs the definite integral with piecewise linear technique to calculate the error between the original posynomial and the corresponding ET/PCT underestimators. Lastly, this study aims to identify an indicator that can choose the appropriate transformation between ET and PCT and analyze the correctness of the proposed indicator for posynomial terms in nonlinear problems. The proposed indicator can efficiently solve nonlinear problems containing posynomial terms. Numerical examples are used to demonstrate the efficacy of the proposed indicator.

A Mixed-Integer Second-Order Cone Programming Algorithm for the Optimal Power Distribution of AC-DC Parallel Transmission Channels

For the controllability of the transmission power of DC transmission channels, the optimal power distribution (OPD) of AC-DC parallel transmission channels is an effective measure for improving the economic operation of an AC-DC interconnected power grid. A dynamic optimal power flow model for day-ahead OPD of AC-DC parallel transmission channels is established in this paper. The power flow equation constraints of an AC-DC interconnected power grid and the constraints of the discrete regulation requirement of the transmission power of DC channels are considered, which make the OPD model of the AC-DC parallel transmission channels a mixed-integer nonlinear non-convex programming (MINNP) model. Through a cone relaxation transformation and a big M method equivalent transformation, the non-convex terms in the objective function and constraints are executed with the convex relaxation, and the MINNP model is transformed into a mixed-integer second-order cone programming model that can be solved reliably and efficiently using the mature optimization solver GUROBI. Taking an actual large-scale AC-DC interconnected power grid as an example, the results show that the OPD scheme of the AC-DC parallel transmission channels obtained by the proposed algorithm can effectively improve the economical operation of an AC-DC interconnected power grid.

Probability-driven transmission expansion planning with high-penetration renewable power generation: A case study in northwestern China

The incorporation of power generation derived from renewable energy sources, or renewable power generation (RPG), into conventional electric power grids has been rapidly increasing on a large scale in recent years. However, this process can be expected to inevitably increase the uncertainty associated with transmission line investment owing to the inherent uncertainty associated with RPG. While robust transmission expansion planning (RTEP) has been commonly employed for optimizing transmission line investments, this method suffers from serious disadvantages such as the neglect of available RPG probability information, overly conservative solutions, and exceedingly time consuming solution processes. The present work addresses these disadvantages by modeling the probability of RPG uncertainty according to RPG output probabilities obtained over a long-term planning horizon based on available historical data using a hybrid probability uncertainty set constructed using 1-norm and ∞-norm metrics. A probability-driven RTEP model is then proposed to obtain an optimal investment strategy that provides a security guarantee under the worst-case RPG probability distribution, while alleviating the need for overly conservative solutions with high total expansion costs. In addition, a modified column-and-constraint generation algorithm is developed to solve the proposed tri-level probability-driven RTEP model, where the large-scale inner bi-level component of the optimization problem is decomposed into several small-scale linear models that can be solved in parallel. The proposed algorithm requires no dual variables in the inner problem and eliminates all highly non-convex bilinear terms like those obtained in conventional RTEP solution algorithms. This can effectively increase the computational speed of the solution process. The effectiveness, good applicability, and robustness of the proposed model and solution algorithm are demonstrated by numerical applications based on a Garver’s 6-bus test system and an existing electric grid system in northwestern China.

The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach

In this paper, we establish some new results for the existence and nonexistence of radial large positive solutions for the modified Schrödinger system with a nonconvex diffusion term. Our main tools are the successive iteration technique and dual approach. A necessary and sufficient condition for the existence of radial large positive solutions is also given, which improves and extends many previous work in this field of research.

An alternative efficient representation for the project portfolio selection problem

Project portfolio selection problem (PPSP) is usually formulated as a mixed integer polynomial program with cross-product terms. The problem is hard to solve due to the non-convex cross-product terms involved. To find an exact optimal solution, currently available methods adopt different linearization techniques to handle the cross-product terms and then utilize a branch-and-bound scheme for computations. This study proposes an alternative efficient representation for PPSP using fewer continuous variables than the current methods to achieve global optimum. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed method. In addition, the proposed method is integrated with a general binary cut scheme for identifying all alternative solutions for decision makers to consider better options.

A trust region approach for numerical modeling of non-isothermal phase change

An efficient enthalpy-based numerical approach is introduced for solving heat transfer problems with non-isothermal phase change. The goal of this model is to simulate permafrost thaw rates, understand the significant effects of climate change on permafrost, and assess the reciprocal effects of permafrost thaw on climate. In order to simulate permafrost over time spans of several years, a robust and efficient model is required. In the present setting, the heat transfer problem is converted to a minimization problem, in which we minimize a potential function that characterizes the governing heat transfer PDE within a time discrete framework. The use of the trust region minimization algorithm proves desirable due to the highly non-linear energy functional which also involves non-convex terms induced by phase change. Results obtained show satisfactory agreement with existing analytical solutions. Moreover, the grid and time step convergence studies conducted to examine the dependence of the solution on mesh and time step sizes indicate robust convergence rates. This is the first application of trust region energy minimization algorithm in permafrost simulation.

A bilevel decomposition method for the simultaneous heat integration and synthesis of steam/organic Rankine cycles

This work tackles the simultaneous synthesis and design of heat exchanger networks (HEN) integrated with complex utility systems, such as Heat Recovery Steam Cycles or Organic Rankine Cycles. Thanks to the combination of two superstructures (Rankine cycles and HEN), all the key heat integration options between process and utility system can be considered, and the trade-off between efficiency and costs is optimized. The superstructure for complex utility systems involves streams with variable flow rate. The resulting MINLP problem is very challenging due to its large number of binary variables and nonconvex terms. We present a novel bilevel decomposition algorithm, combining the outer-approximation linearization technique with McCormick relaxation, valid redundant constraints, piecewise linearization of cost functions and “nested” integer cuts. The algorithm successfully tackled real-world problems with up to 35 streams showing considerable improvements in solution quality and computational time over commercial MINLP solvers and meta-heuristic algorithms.

Representing ZIP loads in convex relaxations of optimal power flow problems

The ZIP load models consist of constant-impedance, constant-current, and constant-power loads, which is crucial to access the network performance. Generally, including the constant-impedance and constant-power load model in the optimal power flow (OPF) problem is typically straightforward in convex formulations. However, further inclusion of an exact constant-current load model results in non-convex terms and is computationally challenging. Hence, this paper proposes a new approach for extending convex relaxations of OPF problems to include representations of ZIP load models. Exploiting knowledge of the reference bus’ phase angle, this approach uses geometric relationships to relax the voltage magnitude expressions in order to model the constant-current components, thus enabling convex representations of ZIP load models. The proposed approach is applied to semidefinite, second-order cone and quadratic convex programming relaxations of the OPF problem and demonstrated using several test cases.

Extended opportunity cost model to find near equilibrium electricity prices under non-convexities

This paper finds near equilibrium prices for electricity markets with non-convexities due to binary variables, in order to reduce the market participants’ opportunity costs, such as generators’ unrecovered costs. The opportunity cost is defined as the difference between the profit when the instructions of the market operator are followed and when the market participants can freely make their own decisions based on the market prices. We use the minimum complementarity approximation to the minimum total opportunity cost model, from previous research, with tests on a much more realistic unit commitment model than in previous research, including features such as reserve requirements, ramping constraints, and minimum-up and -down times. The developed model incorporates flexible price-responsive demand, as in previous research, but since not all demand is price responsive, we consider the more realistic case that total demand is a mixture of fixed and flexible. Another improvement over previous minimum total opportunity cost research is computational: whereas the previous research had nonconvex terms among the objective function’s continuous variables, we convert the objective to an equivalent form that contains only linear and convex quadratic terms in the continuous variables, thus allowing for efficient optimization by CPLEX.We compare the unit commitment model with the standard social welfare optimization version of unit commitment, in a series of sensitivity analyses, varying flexible demand to represent varying degrees of future penetration of electric vehicles and smart appliances, different ratios of generation availability, and different values of transmission line capacities to consider possible congestions. The minimum total opportunity cost and social welfare solutions are mostly very close in different scenarios, except in some extreme cases; the obtained solution has smaller opportunity costs than social welfare, at very little cost in reduction of social welfare; and solution times of the minimum total opportunity cost model are longer, due to a larger model size, but the times are still very quick for practical use.

Convex optimization based adjustable robust dispatch for integrated electric-gas systems considering gas delivery priority

With the mushrooming deployment of volatile renewable energy sources as well as the intrinsic uncertainty from the demand side, secure and economic energy dispatch has become increasingly challenging for energy systems, especially for the emerging and promising integrated electric-gas system. Aside from the aforementioned uncertainties, the dispatch of the integrated electric-gas system suffers from two inherent obstacles, which are model nonconvexities, originating from the Weymouth equations in the gas network, and the demand-side differentiated gas delivery priorities according to current industrial practice, respectively. To deal with the conundrum, an adjustable robust dispatch method is proposed for operating the integrated electric-gas system, where uncertain wind generation outputs and gas loads are described by intervals. In contrast to existing work employing pre-determined intervals, the admissible wind output intervals in this paper are optimized, reflecting the interdependencies between the regulation capabilities of gas-fired generation and the gas delivery adequacy. By this means, gas delivery priority is considered in comply with gas industrial practice, and it also provides a more flexible mechanism to maintain robustness of the dispatch strategy. Through analyzing the feasibility impact of uncertain variables, a deterministic robust counterpart is derived, in which uncertainties are eliminated based on affine generator dispatch and estimation of total line pack. Furthermore, nonconvex quadratic terms in the Weymouth equations are expressed as difference-of-convex functions. A sequential convex optimization procedure is developed, and a heuristic method is suggested to initiate the sequential algorithm. The proposed models and methods are tested on three test systems. Key impact factors on the dispatch strategy, such as gas prices, wind generation forecast accuracy and gas network initial operation conditions, are analyzed, and the computational benefits brought by convex programming are validated by scalability tests.

A new nonconvex approach for image restoration with Gamma noise

In this paper, a new total generalized variational(TGV) model for restoring images with multiplicative noise is proposed, which contains a nonconvex fidelity term and a TGV term. We use a difference of convex functions algorithm (DCA) to deal with the proposed model. For multiplicative noise removal, there exist many models and algorithms, most of which focus on convex approximation so that numerical algorithms with guaranteed convergence can be designed. Unlike these algorithms, we use the DCA algorithm to remove multiplicative noise. By numerical experiments, it is shown that the proposed approach leads to a better solution compared with the gradient projection algorithm for solving the classic multiplicative noise removal models. We prove that the sequence generated by the DCA algorithm converges to a stationary point, which satisfies the first order optimality condition. Finally, we demonstrate the performance of our whole scheme by numerical examples. A comparison with other methods is provided as well. Numerical results demonstrate that the proposed algorithm significantly outperforms some previous methods for multiplicative Gamma noise removal.

Robust Sparse Recovery in Impulsive Noise via M-Estimator and Non-Convex Regularization

Robust sparse recovery aims at recovering a sparse signal or image from its compressed and contaminated measurements. Under the impulsive noise condition, the performance of traditional sparse recovery algorithms may deteriorate seriously for exploiting l 2 -norm to model the non-Gaussian noise. In this paper, a novel formulation which combines the M-estimator and the non-convex regularization term is presented to address the issue of robust sparse recovery in the impulsive noise environment. Since the l 2 -norm is highly sensitive to the large outliers appearing in impulse interference, we replace it with the robust M-estimators that have exhibited the powerful capability of suppressing impulsive noise in various scenarios. Meanwhile, the non-convex regularization is capable of overcoming the biased estimation problem induced by the convex l 1 -norm regularization and thus can obtain more accurate reconstruction results. Furthermore, to solve the resulting non-convex formulation, an efficient first-order algorithm with low computational complexity is developed by utilizing the alternating direction method of multipliers framework and the half-quadratic optimization. The reconstruction experiments under the circumstance of impulsive noise are conducted to demonstrate the superior performance of the proposed algorithm over several typical sparse recovery algorithms.

A Stochastic Semismooth Newton Method for Nonsmooth Nonconvex Optimization

In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via calling stochastic first and second order oracles. The proposed method can be seen as a hybrid approach combining stochastic semismooth Newton steps and stochastic proximal gradient steps. Two inexact growth conditions are incorporated to monitor the convergence and the acceptance of the semismooth Newton steps and it is shown that the algorithm converges globally to stationary points in expectation and almost surely. We present numerical results and comparisons on l1-regularized logistic regression and nonconvex binary classification that demonstrate the efficiency of the algorithm.

Iterative Adaptive Nonconvex Low-Rank Tensor Approximation to Image Restoration Based on ADMM

In this paper, in order to recover more finer details of the image and to avoid the loss of image structure information for image restoration problem, we develop an iterative adaptive weighted core tensor thresholding (IAWCTT) approach based on the alternating direction method of multipliers (ADMM). By observing the decoupling property of the ADMM algorithm, we first propose that the key step to image restoration is to tackle the denoising subproblem efficiently using appropriate prior information. Secondly, by analyzing the properties of the core tensor, we propose that low-rank tensor approximation can be implemented by penalizing the core tensor itself, instead of penalizing the CP rank, Tucker rank or the multilinear rank and Tubal rank. The IAWCTT approach is proposed to solve the denoising subproblem in the ADMM framework, and we claim that such an adaptive weighted scheme is equivalent to a kind of nonconvex penalty for the core tensor; thus, it is unnecessary to use the nonconvex penalty term to induce strong sparse/low-rank solution in image restoration optimization problem, because the scheme that selecting appropriate weights to the convex penalty term can also lead to strong sparse/low-rank solution. Numerical experiments show that our proposed model and algorithm are comparable to other state-of-the-art models and methods.