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Fourier phase retrieval is one of the representative inverse problems where a signal needs to be recovered using only the measured magnitude of its Fourier transform. Deep learning-based algorithms for solving Fourier phase retrieval have been widely studied. These methods provide better reconstruction than the conventional algorithms, such as alternating projection approaches and convex relaxation methods. However, it is difficult to recover the phase information of 256 × 256 images accurately, and they often cannot provide fine details and textures. Recently, diffusion models have been used to solve Fourier phase retrieval problems. They offer realistic reconstruction results, but due to the nature of generative models, they often create non-existent features in the actual images. To address these issues, we introduced a novel algorithm inspired by regularization by denoising diffusion, a variational diffusion sampling for reconstructing the images from the measurements. In particular, the optimization problem in the convex relaxation approach for phase retrieval is interpreted as an additional constraint during the variational sampling process to estimate the phase from the given Fourier magnitude measurement. The proposed method stands out by leveraging not only pre-trained diffusion models as image priors but also the classical optimization approach as the regularization. This novel combination ensures not just accurate phase reconstruction, but also performance guarantees. Our experiments demonstrate that the proposed algorithm consistently provides state-of-the-art performance across various datasets of 256 × 256 images. We further showed the effectiveness of the new regularization for the performance gain in the phase estimation. [Display omitted] • Introduces a novel PhaseCut regularization technique based on convex relaxation of phase for Fourier phase retrieval problem. • Adopts a variational sampler using pre-trained diffusion models for Fourier phase retrieval problem as image priors. • Achieves the state-of-the-art performance with various datasets of 256 × 256 images. [ABSTRACT FROM AUTHOR]

Different from most studies only incorporating economic aspect into operation scheduling, this paper develops a novel network-constrained framework that can address both economic and secure issues for combined heating and power (CHP) microgrids integrated with wind power considering uncertainties. According to quadratically constrained programs, a two-stage network-constrained stochastic programming (TSNCSP) model is formulated to improve economic benefits and meanwhile capture active/reactive power and off-nominal bus voltage constraints in the full decision variable domain. In the first stage, namely day-ahead, the schedule of transacted electricity with the upstream power grid and the generation cost of heat is determined according to the forecast information. In the second stage, namely real-time, a recourse function of adjustable resources is defined to reduce the expected cost incurred by the perturbation of random wind power outputs. Moreover, the original non-convex problem is innovatively transformed into a semidefinite programming through incorporation of demand response program (DRP), duality, complementary slackness, and relaxation techniques to improve the solving efficiency. Finally, a proposition is presented and proved that provides a sufficient condition for the exactness of the proposed convex relaxation. Numerical simulations on the 33-bus test system verify the effectiveness of the proposed framework in multi-period scenario-based scheduling problems. • Two-stage network-constrained stochastic programming schedules a CHP microgrid. • A perturbed problem is proposed based on duality and complementarity slackness. • Convex relaxation of the problem offers the advantage of global optimum. • Wind power forecasting errors are accommodated considering network constraints. • A proposition presents sufficient conditions for the exact convex relaxation. [ABSTRACT FROM AUTHOR]

This article proposes an efficient global algorithm for solving general linear multiplicative programming problem (GLMP). The new algorithm combines the quadratic convex relaxation problem, adaptive branching rule, region reduction technique and branch-and-bound scheme. Firstly, a transformation technique can transform GLMP into a non-convex quadratic program with quadratic and linear constraints. The non-convexity parts of the equivalent problem are addressed by employing secant lines, so that a quadratic convex relaxation problem is structured. Secondly, we introduce the adaptive branching rule to improve the upper bound of the optimal value. Thirdly, the convergence of the proposed algorithm is analyzed and its worst-case complexity is provided. Finally, numerical experiments demonstrate the efficiency and advantage of the proposed algorithm for obtaining the global ɛ -optimal solutions of test instances. • A novel branch-and-bound algorithm is presented for globally solving GLMP. • The algorithm integrates QCR, adaptive branching rule, region deletion technique and BB scheme. • The global convergence and computational complexity of the algorithm are reported. • Numerical results show the effectiveness of the algorithm to obtain an optimal solution of GLMP. [ABSTRACT FROM AUTHOR]

In recent years, there has been significant interest in the development of machine learning-based optimization proxies for AC Optimal Power Flow (AC-OPF). Although significant progress has been achieved in predicting high-quality primal solutions, no existing learning-based approach can provide valid dual bounds for AC-OPF. This paper addresses this gap by training optimization proxies for a convex relaxation of AC-OPF. Namely, the paper considers a second-order cone (SOC) relaxation of AC-OPF, and proposes a novel architecture that embeds a fast, differentiable (dual) feasibility recovery, thus providing valid dual bounds. The paper combines this new architecture with a self-supervised learning scheme, which alleviates the need for costly training data generation. Extensive numerical experiments on medium- and large-scale power grids demonstrate the efficiency and scalability of the proposed methodology. • Dual conic proxy (DCP) is proposed for convex relaxation of AC-OPF. • A novel, dual-feasible machine learning architecture is presented. • The model is trained end-to-end using self-supervised learning. • DCP achieves < 2% gap and a 1000x speedup compared to Mosek. • DCP provides valid dual bounds to verify optimality of AC-OPF solutions. [ABSTRACT FROM AUTHOR]

AC State Estimation (ACSE) is widely recognized as a practical approach for determining the grid states in steady-state conditions. It serves as a fundamental analysis to ensure grid security and is a reference for market dispatch. As grid complexity increases with rapid electrification and decarbonization, there is a growing need for more accurate knowledge of the grid operating state. However, existing ACSE algorithms have technical gaps. Critically, current ACSE algorithms are susceptible to erroneous system parameters, which are assumed to be fixed in traditional approaches. In this paper, we build a novel circuit-theoretic joint parameter-state estimation algorithm to address this limitation. The innovative algorithm builds an analogous equivalent circuit of the grid with states and certain parameters unknown. It solves a circuit-constrained optimization to estimate the most likely grid states and parameters given a set of measurements. Further, it quantifies the goodness of the estimated output by formulating tight convex envelopes around the original non-convex problem to quantify the quality of estimates. We compare the various proposed approaches on systems with up to 2869 nodes while demonstrating a tradeoff between solution optimality and model fidelity. • Proposing a novel circuit-theoretic framework for joint parameter- state estimation. • Experiments recover unknown states and parameters with a quantifiable level of accuracy. • Proposing enhanced convex relaxations, improve on McCormick approach. • Proposed method balances solution optimality & network AC feasibility. [ABSTRACT FROM AUTHOR]

The increasing deployment of energy storages in power grid necessitates the consideration of their operational costs and constraints. However, energy storages introduce complementary constraints or binary variables, make the optimization problems non-convex and challenging to solve. To tackle this issue, we propose a generalized relaxation condition (GRC), which possesses a priori property. As long as the GRC is met, it is guaranteed that there is no simultaneous charging and discharging (SCD) in the relaxed optimization problems excluding the complementarity constraints. Moreover, we prove that the exact relaxation region on the locational marginal prices (LMPs) formed by the proposed GRC contains those of the other existing relaxation conditions. The proposed GRC can be satisfied for the majority of electricity price scenarios, including positive prices and mild negative prices. Beyond that, through introducing penalties for charging and discharging losses of storage into the objective function, the proposed GRC can be further ensured in all price scenarios. We also develop a criterion to set the penalty parameter properly, making the GRC satisfied. Afterward, a practical relaxation method is presented based on the proposed GRC. Finally, we propose a practical two-stage penalty parameter adjustment scheme to weaken the dependence of the proposed relaxation method on the prediction accuracy of the LMPs. The proposed exact relaxation condition is applicable to various scenarios, such as storage-participating economic dispatch, joint energy-reserve optimization and Volt/VAR optimization. Finally, comprehensive test cases verify the exactness and advantages of the proposed relaxation method. • An exact generalized relaxation condition (GRC) with a priori property is proposed for complementarity constraints of energy storages. • The proposed GRC has a broader scope than the existing relaxation conditions and is applicable to various power system optimization problems. • By introducing penalties for charging and discharging losses of storage into the objective, a practical relaxation method is developed based on the proposed GRC. • The applicability of the proposed GRC can be ensured for scenarios with arbitrary electricity prices by adjusting the penalty parameter. [ABSTRACT FROM AUTHOR]

We present a convex cone program to infer the latent probability matrix of a random dot product graph (RDPG). The optimization problem maximizes the Bernoulli maximum likelihood function with an added nuclear norm regularization term. The dual problem has a particularly nice form, related to the well-known semidefinite program relaxation of the maxcut problem. Using the primal-dual optimality conditions, we bound the entries and rank of the primal and dual solutions. Furthermore, we bound the optimal objective value and prove asymptotic consistency of the probability estimates of a slightly modified model under mild technical assumptions. Our experiments on synthetic RDPGs not only recover natural clusters, but also reveal the underlying low-dimensional geometry of the original data. We also demonstrate that the method recovers latent structure in the Karate Club Graph and synthetic U.S. Senate vote graphs and is scalable to graphs with up to a few hundred nodes. [ABSTRACT FROM AUTHOR]

This work establishes and validates a Grid Graph Signal Processing (G-GSP) framework for estimating the state vector of a radial distribution feeder. One of the key insights from GSP is the generalization of Shannon’s sampling theorem for signals defined over the irregular support of a graph, such as the power grid. Using a GSP interpretation of Ohm’s law, we show that the system state can be well approximated with relatively few components that correspond to low-pass Graph Fourier Transform (GFT) frequencies. The target application of this theory is the formulation of a three-phase unbalanced Distribution System State Estimation (DSSE) formulation that recovers the GFT approximation of the system state vector from sparse Advanced Metering Infrastructure (AMI) measurements. To ensure convergence of G-GSP for DSSE, the proposed solution relies on a convex relaxation technique. Furthermore, we propose an optimal sensor placement algorithm for AMI measurements. Numerical results demonstrate the efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

Distributed secondary control stands out for its flexibility and expandability in microgrids (MGs) control, in where communication network plays a fundamental and critical role. The communication topology and link-weights have significant impact on the attack-resilience and dynamic performance of MGs, which should be appropriately designed. However, the joint optimization problem of communication topology and link-weights in the existing literature of consensus-based secondary voltage control of MGs is usually neglected. To bridge this gap, we propose a novel two-stage optimization approach for the communication network design, which jointly optimizes the topological structure to enhance the structural survivability and link weights to improve the dynamic performance (including the speed of convergence and robustness to time-delay). The first stage problem is formulated into a mixed-integer semi-define programming (MISDP) model based on convex relaxation technique, which is then converted equivalently into an integer quadratic programming (IQP) problem and then a MISDP feasibility problem to facilitate the solution. The second stage problem is formulated into a bi-objective SDP model to compromise between the convergence performance and robustness to time-delay. Simulations based on a microgrid with 10 distributed generation (DG) units under different scenarios are implemented to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Flexible transmission line impedances on one hand are a promising control resource for facilitating grid flexibility, but on the other hand add much complexity to the concerned optimization problems. This paper develops a convexification method for the AC optimal power flow with flexible line impedances. First, it is discovered that a flexible-impedance line is equivalent to a constant-impedance line linking a pair of transformers with correlated and continuously adjustable tap ratios. Then, with this circuit equivalent, the original optimization problem is reformulated into a semi-definite program under the existing convex relaxation framework, which improves the solution tractability and optimality in an easy-to-implement manner. The proposed method is verified by numerical tests on the IEEE 118-bus system. [ABSTRACT FROM AUTHOR]

Natural gas-fired generation units can hedge against the volatility in the uncertain renewable generation, which may occur during very short periods. It is crucial to utilize models capable of correctly capturing the natural gas network dynamics induced by the volatile demand of gas-fired units. The Weymouth equation is commonly implemented in literature to avoid dealing with the mathematical complications of solving the original governing differential equations of the natural gas dynamics. However, it is shown in this paper that this approach is not reliable in the short-term operation problem. Here, the merit of the non-convex transient model is compared with the simplified Weymouth equation, and the drawbacks of employing the Weymouth equation are illustrated. The results demonstrate how changes in the natural gas demand are met by adjustment in the pressure within pipelines rather than the output of natural gas suppliers. This work presents a convex relaxation scheme for the original non-linear and non-convex natural gas flow equations with dynamics, utilizing a rank minimization approach to ensure the tightness. The proposed method renders a computationally efficient framework that can accurately solve the non-convex non-linear gas operation problem and accurately capture its dynamics. Also, the results suggest that the proposed model improves the solution optimality and solution time compared to the original non-linear non-convex model. Finally, the scalability of the proposed approach is verified in the case study. [ABSTRACT FROM AUTHOR]

It is common to have to process signals or images whose values are cyclic and can be represented as points on the complex circle, like wrapped phases, angles, orientations, or color hues. We consider a Tikhonov-type regularization model to smoothen or interpolate circle-valued signals defined on arbitrary graphs. We propose a convex relaxation of this nonconvex problem as a semidefinite program, and an efficient algorithm to solve it. [ABSTRACT FROM AUTHOR]

Combined heat and electricity operation with variable mass flow rates promotes flexibility, economy, and sustainability through synergies between electric power systems (EPSs) and district heating systems (DHSs). Such combined operation presents a highly nonlinear and nonconvex optimization problem, mainly due to the bilinear terms in the heat flow model—that is, the product of the mass flow rate and the nodal temperature. Existing methods, such as nonlinear optimization, generalized Benders decomposition, and convex relaxation, still present challenges in achieving a satisfactory performance in terms of solution quality and computational efficiency. To resolve this problem, we herein first reformulate the district heating network model through an equivalent transformation and variable substitution. The reformulated model has only one set of nonconvex constraints with reduced bilinear terms, and the remaining constraints are linear. Such a reformulation not only ensures optimality, but also accelerates the solving process. To relax the remaining bilinear constraints, we then apply McCormick envelopes and obtain an objective lower bound of the reformulated model. To improve the quality of the McCormick relaxation, we employ a piecewise McCormick technique that partitions the domain of one of the variables of the bilinear terms into several disjoint regions in order to derive strengthened lower and upper bounds of the partitioned variables. We propose a heuristic tightening method to further constrict the strengthened bounds derived from the piecewise McCormick technique and recover a nearby feasible solution. Case studies show that, compared with the interior point method and the method implemented in a global bilinear solver, the proposed tightening McCormick method quickly solves the heat–electricity operation problem with an acceptable feasibility check and optimality.

Regarding the nonconvexity of the optimal power flow (OPF) problem, power industry typically employs convex approximation approaches to satisfy the computational demand. However, the accuracy performances of the convex OPF models are hard to be predicted in actual operating power systems. To control the maximum effect of the approximation error on power dispatch, it is important to solve and compare the worst-case approximation errors of these OPF models (i.e., the error bounds). In this paper, the error bound is represented by the maximum gap of the optimal solutions of the nonconvex and convex OPF models (i.e., the optimality gap) which is further equivalent to be the Hausdorff distance of the feasible regions of the two OPF models. Subsequently, an optimization model is proposed to generally solve the Hausdorff distance as well as the error bound. Case studies show that the proposed error bound is tight and effective when the power system is not seriously congested. The relaxation rate of the error bounds is less than 5% when congestion rate is less than 10%. It provides theoretical guidance for selecting the convex OPF model under the condition without historical operating data. • The error bound of convex OPF models is generally modeled based on Fenchel duality. • An approach for solving error bounds is proposed from Hausdorff distance calculation. • The accuracy and application of the error bound measurement are numerically presented. [ABSTRACT FROM AUTHOR]

This study considers optimization problems for time-varying network topologies that maximize the controllability of linear network systems under sparsity constraints. The major theoretical differences from the existing findings on control node scheduling problems is that our method is based on the matrix-valued Pontryagin maximum principle applied to the controllability Lyapunov differential equation. This allows us to handle various controllability metrics that can only be approximated by existing methods. Finally, we perform a numerical evaluation of our proposed method and compare results with those of our previous approximation method. [ABSTRACT FROM AUTHOR]

Due to the increasing penetration of large-scale renewable energy generation at transmission level, hybrid AC/DC transmission expansion is becoming more relevant by the day, especially if the considered renewable generation is far away from existing AC grid infrastructure. This paper presents a multi-period hybrid AC/DC network expansion planning model that incorporates node voltages, reactive power and network losses. Since the operation of a high voltage DC link is influenced considerably by the converter stations, converter transformer, filter, and phase reactor are modeled in each converter station. Since the AC power flow equations render a non-convex problem whose solution algorithms do not always guarantee convergence or optimality, a second-order conic relaxation is utilized. The resulting model is a mixed-integer convex problem with a convergence guarantee that embodies a better approximation of system operation than its DC linear counterpart. Three case studies are considered to illustrate the proposed model. [ABSTRACT FROM AUTHOR]

The evolution of power distribution grids from passive to active systems creates reliability and efficiency challenges to the distribution system operators. In this paper, an energy management and control scheme for managing the operation of an active distribution grid with prosumers is proposed. A multi-objective optimization model to minimize (i) the prosumers electricity cost and (ii) the cost of the grid energy losses, while guaranteeing safe and reliable grid operation is formulated. This is done by determining the active and reactive power set-points of the photovoltaic and storage systems integrated in the grid buildings. The resulting optimization model is non-convex, thus a convex second-order cone program is developed by appropriately relaxing the non-convex constraints which yields optimal results in most operating conditions. The convexified model is further utilized to develop an algorithm that yields feasible solutions to the non-convex problem under any operating conditions. Moreover, a second novel algorithm to find the operating point that provides fairness between the prosumers and the grid costs is proposed. Simulation results demonstrate the effectiveness and superiority of the proposed scheme in managing an industrial distribution grid compared to a self-consumption approach. [ABSTRACT FROM AUTHOR]

The knowledge of distribution grid models, including topologies and line impedances, is essential for grid monitoring, control and protection. However, such information is often unavailable, incomplete or outdated. The increasing deployment of smart meters (SMs) provides a unique opportunity to tackle this issue. This paper proposes a two-stage framework for distribution grid modeling using SM data. In the first stage, the network topology is identified by reconstructing a weighted Laplacian matrix of distribution networks. In the second stage, a least absolute deviations (LAD) regression model is developed for estimating line impedance of a single branch based on the nonlinear (inverse) power flow model, wherein a conductor library is leveraged to narrow down the solution space. The LAD regression model is originally a mixed-integer nonlinear program whose continuous relaxation is still non-convex. Thus, we specially address its convex relaxation and discuss the exactness. The modified regression model is then embedded within a bottom-up sweep algorithm to achieve the identification across the network in a branch-wise manner. Numerical results on the IEEE 13-bus, 37-bus and 69-bus test feeders validate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this work, we study complex-valued data detection performance in massive multiple-input multiple-output (MIMO) systems. We focus on the problem of recovering an n-dimensional signal whose entries are drawn from an arbitrary constellation K ⊂ C from m noisy linear measurements, with an independent and identically distributed (i.i.d.) complex Gaussian channel. Since the optimal maximum likelihood (ML) detector is computationally prohibitive for large dimensions, many convex relaxation heuristic methods have been proposed to solve the detection problem. In this paper, we consider a regularized version of this convex relaxation that we call the regularized convex relaxation (RCR) detector and sharply derive asymptotic expressions for its mean square error and symbol error probability. Monte-Carlo simulations are provided to validate the derived analytical results. [ABSTRACT FROM AUTHOR]

A piezoelectric-type tensor is of order three that is symmetric with respect to the last two indices. Its largest C-eigenvalue determines the highest piezoelectric coupling constant. However, computing the largest C-eigenvalue is NP-hard. This paper addresses this problem using convex relaxation. To this end, we first establish an equivalence property that helps to rewrite the problem as a matrix optimization over rank-1 together with the partially symmetric tensor constraints. Then, via Lagrangian dual, convex relaxations are derived, yielding the problems of maximization of a linear function over one or two matrix nuclear norm constraint(s), together with a partially symmetric tensor constraint. Such relaxations define new norms, which are smaller than the spectral norms of the corresponding unfolding matrices. Several insights are provided for the tightness issues of the convex relaxations, including a certification, a sufficient criterion, and an equivalence condition. The spectral property of the dual variable, in particular, determines the tightness. When the convex relaxations are not tight, an approximation algorithm is proposed to extract a feasible approximation solution, with a theoretical lower bound provided. We provide several types of tensors to justify the tightness of the convex relaxations. In case that the relaxations are not tight, their optimal values, serving as upper bounds, are still tighter than those in the literature. [ABSTRACT FROM AUTHOR]

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer's well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation. [ABSTRACT FROM AUTHOR]

We consider a class of infinite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given finite set MRm. Such hybrid discrete-continuous problems occur in, e.g., topology optimization or medical imaging and are challenging due to their lack of weak lower semicontinuity. To circumvent this difficulty, we introduce as a regularization term a convex integral functional with an integrand that has a polyhedral epigraph with vertices corresponding to the values of M; similar to the L1 norm in sparse regularization, this "vector multibang penalty" promotes solutions with the desired structure while allowing the use of tools from convex optimization for the analysis as well as the numerical solution of the resulting problem. We show well-posedness of the regularized problem and analyze stability properties of its solution in a general setting. We then illustrate the approach for three specific model optimization problems of broader interest: optimal control of the Bloch equation, optimal control of an elastic deformation, and a multimaterial branched transport problem. In the first two cases, we derive explicit characterizations of the penalty and its generalized derivatives for a concrete class of sets\scrM. For the third case, we discuss the algorithmic computation of these derivatives for general sets. These derivatives are then used in a superlinearly convergent semismooth Newton method applied to a sequence of regularized optimization problems. We illustrate the behavior of this approach for the three model problems with numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses a target localization problem in 3-D wireless sensor networks using a hybrid system that fuses received signal strength and angle of arrival measurements. First, we formulate the received signal strength and angle of arrival measurement models as the pseudo-linear equations. Then, the bias is derived from the 3-D angle of arrival measurements that take the measurement noise into account to improve the localization performance. Furthermore, a non-convex estimator is derived based on the Least Squares criterion. Finally, semi-definite relaxation and second-order cone relaxation are applied to transform the derived non-convex estimator into a convex one. We propose a semi-definite relaxation and second-order cone relaxation-based estimator which yields the best performance under a large measurement noise or a small measurement noise. The generalization of the proposed method for known transmit power can also be applied to the case when transmit power is not know. Theoretical analysis and computer simulations corroborate the superior performance of the proposed localization methods over the existing ones.

In this article, a novel second-order cone programming based branch flow model is proposed for optimal power flow (OPF) in radial distribution networks. The main advantage of the proposed architecture is that it can recover the bus voltage angle difference and, thus, control the reactive power flow (PF), leading to a better voltage regulation in the network. Based on the line PF and the line impedance, a convex equation is derived and included with the model to retrieve the bus voltage angle difference. Finally, using the retrieved bus voltage angle difference within the global optimal solution space, an algorithm is developed to control the reactive flow for voltage regulation. The proposed model has been tested on modified IEEE 32 bus (IEEE 33 bus network) and IEEE 123 bus power network systems for different test cases. The simulation results are verified with the OpenDSS PF and the MatPower nonlinear programming (NLP) based OPF solutions. The computational efficiency of the proposed model is superior to the NLP models. Also, the proposed reactive flow control algorithm improves the voltage regulation of a network and promises a globally optimal solution. [ABSTRACT FROM AUTHOR]

In this letter, we propose a data-driven warm start approach, empowered by an artificial neural network, to boost the efficiency of convex relaxations in optimal gas flow. Case studies show that this approach significantly decreases the number of iterations for the convex-concave procedure algorithm, while optimality and feasibility of the solution can still be guaranteed. We also confirm the robustness of this algorithm, and show that this approach can be extended to the optimal dispatch of large-scale electricity-gas coupled integrated energy systems. [ABSTRACT FROM AUTHOR]

The optimization problem with sparsity arises in many areas of science and engineering such as compressed sensing, image processing, statistical learning and data sparse approximation. In this paper, we study the dual-density-based reweighted ℓ 1 -algorithms for a class of ℓ 0 -minimization models which can be used to model a wide range of practical problems. This class of algorithms is based on certain convex relaxations of the reformulation of the underlying ℓ 0 -minimization model. Such a reformulation is a special bilevel optimization problem which, in theory, is equivalent to the underlying ℓ 0 -minimization problem under the assumption of strict complementarity. Some basic properties of these algorithms are discussed, and numerical experiments have been carried out to demonstrate the efficiency of the proposed algorithms. Comparison of numerical performances of the proposed methods and the classic reweighted ℓ 1 -algorithms has also been made in this paper. [ABSTRACT FROM AUTHOR]

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in ℝn. Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 2018, 6, 6307–6332], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for n ≥ 3. [ABSTRACT FROM AUTHOR]

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in ℝn. Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 2018, 6, 6307–6332], we provide a variational approximation through Ginzburg–Landau type energies proving a Γ-convergence result for n ≥ 3. [ABSTRACT FROM AUTHOR]

We study extensions of compressive sensing and low rank matrix recovery to the recovery of tensors of low rank from incomplete linear information. While the reconstruction of low rank matrices via nuclear norm minimization is rather well-understand by now, almost no theory is available so far for the extension to higher order tensors due to various theoretical and computational difficulties arising for tensor decompositions. In fact, nuclear norm minimization for matrix recovery is a tractable convex relaxation approach, but the extension of the nuclear norm to tensors is in general NP-hard to compute. In this article, we introduce convex relaxations of the tensor nuclear norm which are computable in polynomial time via semidefinite programming. Our approach is based on theta bodies, a concept from real computational algebraic geometry which is similar to the one of the better known Lasserre relaxations. We introduce polynomial ideals which are generated by the second-order minors corresponding to different matricizations of the tensor (where the tensor entries are treated as variables) such that the nuclear norm ball is the convex hull of the algebraic variety of the ideal. The theta body of order k for such an ideal generates a new norm which we call the θk-norm. We show that in the matrix case, these norms reduce to the standard nuclear norm. For tensors of order three or higher however, we indeed obtain new norms. The sequence of the corresponding unit-θk-norm balls converges asymptotically to the unit tensor nuclear norm ball. By providing the Gröbner basis for the ideals, we explicitly give semidefinite programs for the computation of the θk-norm and for the minimization of the θk-norm under an affine constraint. Finally, numerical experiments for order-three tensor recovery via θ1-norm minimization suggest that our approach successfully reconstructs tensors of low rank from incomplete linear (random) measurements. [ABSTRACT FROM AUTHOR]

In this paper we consider high-dimensional multiclass classification by sparse multinomial logistic regression. We propose first a feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the nonasymptotic bounds for misclassification excess risk of the resulting classifier. We establish also their tightness by deriving the corresponding minimax lower bounds. In particular, we show that there is a phase transition between small and large number of classes. The bounds can be reduced under the additional low noise condition. To find a penalized maximum likelihood solution with a complexity penalty requires, however, a combinatorial search over all possible models. To design a feature selection procedure computationally feasible for high-dimensional data, we propose multinomial logistic group Lasso and Slope classifiers and show that they also achieve the minimax order. [ABSTRACT FROM AUTHOR]

Mathematical models representing the behavior of electrical loads are an important part of any optimal power flow problem. The current state-of-the-art in unbalanced optimal power flow mostly considers wye-connected, constant power loads. However, for applications such as conservation voltage reduction, it is crucial to model how the consumption of the loads depends on the voltage of the network. This article develops a unified framework to handle a wide variety of load types: delta- or wye-connected, constant power, constant current and exponential load models. Furthermore, it proposes a novel convex relaxation for the exponential model, using power cones, that is intersected next with a well-known semi-definite relaxation of unbalanced OPF. Finally, numerical results on the LVTestCase feeder are included for both the exact, non-linear equations and the convex relaxation, which show how considering the voltage sensitivity and connection type can lead to different objective values and voltage profiles. Additional results for 128 low-voltage networks, examine the quality of the solutions obtained with the proposed relaxation. [ABSTRACT FROM AUTHOR]

This work presents a distributed robust operation framework for the microgrids within the power distribution network. It addresses the uncertainties in photovoltaic generation, as well as active, and reactive loads in the network, by modeling them using polyhedral sets. The energy management problems are formulated as two-stage optimization problems in microgrids and power distribution networks for which Benders decomposition is used to solve the problems. The power flow constraints are relaxed using second-order cone relaxation and moment relaxation techniques in the power distribution network and microgrid energy management problems, respectively. The interaction between the microgrids and the power distribution network is represented using the price signal and the exchanged active and reactive power at the microgrids' points of common coupling. The effectiveness of the proposed framework is shown using two case studies on IEEE 37-bus and IEEE 123-bus systems. The effect of the energy storage system on the operational cost of the system is assessed. It is shown that total operation cost is decreased by 9.69% when the energy storage system is integrated into the power distribution network. Additionally, the solution derived from the robust problem is compared with that of the deterministic problem. • The energy management problems are modeled as two-stage robust optimization problems. • The formulated problems are solved using the Benders decomposition technique. • Two convex relaxation techniques were used to solve the energy management problems in the subproblems. • The coordination between the distribution network and microgrids was ensured using the price signal and power exchanged. [ABSTRACT FROM AUTHOR]

With the integration of renewable energy-based sources and demand response initiatives, the distribution network has transformed into an active network with bidirectional flows. Network Reconfiguration (NR), i.e., changing network topology, is an alternative to minimize power loss in the distribution network with sectionalizing switches (normally closed) and tie-line switches (normally open). Due to the discrete nature of switches (on or off), the NR formulation is generally a mixed-integer NP-hard nonlinear problem. This paper proposes the first NR formulation for an unbalanced active distribution network with continuous variables to indicate switches' statuses in NR while considering mutual coupling between line impedances. The obtained solution provides the optimal real power and voltage magnitude settings of dispatchable Distributed Generators (DGs) and network topology for the minimum active power loss in the network. The proposed formulation considers all practical aspects of the distribution network, i.e., mutual coupling, load unbalances, angular unbalance, and the absence of one or two phases in a line section, which, otherwise, have not been considered together in previous formulations. Further, a novel convex relaxation for bus voltage phase angles avoids explicit voltage recovery. Compared to prior formulations, numeric results for unbalanced radial distribution networks showcase the proposed formulation's effectiveness. • NR for active unbalanced mutually coupled distribution networks is formulated. • The use of continuous switch status variables reduces the problem's complexity. • A novel convex relaxation is used, where bus voltage recovery is not required. • Effect of unbalances and mutual coupling is analyzed on NR and DGs' set points. [ABSTRACT FROM AUTHOR]

Sparse precision matrix estimation, also known as the estimation of the inverse covariance matrix in statistical contexts, represents a critical challenge in numerous multivariate analysis applications. This challenge becomes notably complex when the dimension of the data is far greater than the capacity of samples. To address this issue, we introduce a convex relaxation model that employs the first-order optimality conditions associated with the lasso-penalized D-trace loss for the purpose of estimating sparse precision matrix. The proposed model is effectively solved through the widely recognized alternating direction method of multipliers. Additionally, we provide closed-form solutions to subproblems in each iteration with a computational complexity of O (n p 2) , and establish the convergence of our proposed algorithm. Numerical investigations demonstrate that our algorithm exhibits the capability to handle large-scale datasets and significantly outperforms the existing methods, particularly when dealing with high-dimensional scenarios characterized by a large dimension p. [ABSTRACT FROM AUTHOR]

Nowadays, a large amount of information has to be transmitted or processed. This implies high-power processing, large memory density, and increased energy consumption. In several applications, such as imaging, radar, speech recognition, and data acquisition, the signals involved can be considered sparse or compressive in some domain. The compressive sensing theory could be a proper candidate to deal with these constraints. It can be used to recover sparse or compressive signals with fewer measurements than the traditional methods. Two problems must be addressed by compressive sensing theory: design of the measurement matrix and development of an efficient sparse recovery algorithm. These algorithms are usually classified into three categories: convex relaxation, non-convex optimization techniques, and greedy algorithms. This paper intends to supply a comprehensive study and a state-of-the-art review of these algorithms to researchers who wish to develop and use them. Moreover, a wide range of compressive sensing theory applications is summarized and some open research challenges are presented.

The accuracy of cooperative localization degrades significantly in the non-line-of-sight (NLOS) environments. In addition, the computational complexity of the localization problem often increases dramatically as the scale of a wireless sensor network (WSN) grows. To address these challenges, we propose a distributed NLOS cooperative localization algorithm. First, we propose a new multiplicative model based on the physical mechanism of the NLOS propagation and relax the proposed non-convex model into its convex envelope. This model has a powerful capability to mitigate the NLOS impact and remarkable robustness in changing environments. Second, we design a redundant formulation to decompose the convex problem into numerous sub-problems and then develop an efficient distributed algorithm, which enables each sensor node to locally solve each sub-problem in a parallel way, to decrease the computational complexity. The theoretical analysis and simulations show that the proposed algorithm is superior to the existing methods in both processing speed and localization accuracy.

A solution to the power flow problem is imperative for many power system applications and several iterative approaches are employed to achieve this objective. However, the chance of finding a solution is dependent on the choice of the initial point because of the non- convex feasibility region of this problem. In this paper, a non-iterative approach that leverages a convexified relaxed power flow problem is employed to verify the existence of a feasible solution. To ensure the scalability of the proposed convex relaxation, the problem is formulated as a sparse semi-definite programming problem. The variables associated with each maximal clique within the network form several positive semidefinite matrices. Perturbation and network reconfiguration schemes are employed to improve the tightness of the proposed convex relaxation in order to validate the existence of a feasible solution for the original non-convex problem. Multiple case studies including an ill-conditioned power flow problem are examined to show the effectiveness of the proposed approach to find a feasible solution.