Dual Convex Research Articles

Stochastic dual dynamic programming for optimal power flow problems under uncertainty

Planning in the power sector has to take into account the physical laws of alternating current (AC) power flows as well as uncertainty in the data of the problems, both of which greatly complicate optimal decision making. We propose a computationally tractable framework to solve multi-stage stochastic optimal power flow (OPF) problems in AC power systems. Our approach uses recent results on dual convex semi-definite programming (SDP) relaxations of OPF problems in order to adapt the stochastic dual dynamic programming (SDDP) algorithm for problems with a Markovian structure. We show that the usual SDDP lower bound remains valid and that the algorithm converges to a globally optimal policy of the stochastic AC-OPF problem as long as the SDP relaxations are tight. To test the practical viability of our approach, we set up a case study of a storage siting, sizing, and operations problem. We show that the convex SDP relaxation of the stochastic problem is usually tight and discuss ways to obtain near-optimal physically feasible solutions when this is not the case. The algorithm finds a physically feasible policy with an optimality gap of 3% and yields a significant added value of 27% over a rolling deterministic policy, which leads to overly optimistic policies and underinvestment in flexibility. This suggests that the common industry practice of assuming direct current and deterministic problems should be reevaluated by considering models that incorporate realistic AC flows and stochastic elements in the data as potentially more realistic alternatives.

Optimal Design of Plane Elastic Membranes Using the Convexified Föppl’s Model

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane’s compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified Föppl’s model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain–displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. The primal variables are displacement functions, whilst in the dual one seeks stresses being Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

The volume of an isocanted cube is a determinant*

In any dimension d ≥ 2 , we give exact volume formulas of two mutually polar dual convex d-polytopes. The primal body is called isocanted cube of dimension d, depending on two real parameters 0 < a < ℓ . The limit case a = 0 yields a d-cube of edge-length ℓ. We prove that the volume of such a body is the determinant of the matrix of order d having diagonal entries equal to ℓ and a elsewhere. We also compute the volume of the polar dual body, getting a rational expression in ℓ , a , homogeneous of degree − d with rational coefficients. Isocanted cubes are origin-symmetric zonotopes. Zonoids (defined as the limits of families of zonotopes) satisfy the Mahler conjecture; in particular, zonotopes do. Nonetheless, we confirm (by elementary methods) that the Mahler conjecture holds for isocanted cubes.

Investigating the origin of the far-field reflection interference fringe (RIF) of microdroplets

We show that the reflection interference fringe (RIF) is formed on a screen far away from the microdroplets placed on a prism-based substrate, which have low contact angles and thin droplet heights, caused by the dual convex–concave profile of the droplet, not a pure convex profile. The geometric formulation shows that the interference fringes are caused by the optical path difference when the reflected rays from the upper convex profile at the droplet–air interface interfere with reflection from the lower concave profile at oblique angles lower than the critical angle. Analytic solutions are obtained for the droplet height and the contact angle out of the fringe number and the fringe radius in RIF from the geometric formulation. Furthermore, the ray tracing simulation is conducted using the custom-designed code. The geometric formulation and the ray tracing show excellent agreement with the experimental observation in the relation between the droplet height and the fringe number and the relation between the contact angle and the fringe radius. This study is remarkable as the droplet's dual profile cannot be easily observed with the existing techniques. However, the RIF technique can effectively verify the existence of a dual profile of the microdroplets in a simple setup. In this work, the RIF technique is successfully developed as a new optical diagnostic technique to determine the microdroplet features, such as the dual profile, the height, the contact angle, the inflection point, and the precursor film thickness, by simply measuring the RIF patterns on the far-field screen.

A multi-view ensemble model based on semi-supervised feature learning for small sample classification of PolSAR images

ABSTRACT Classification with a few samples of training set has been a long-standing issue in the field of polarimetric synthetic aperture radar (PolSAR) image analysis and processing. In fact, one of the most important challenges of the PolSAR image classification task is the number of labelled samples. In essence, training network by utilizing just few numbers of training samples cannot be led to the reliable result since the neural network is so sensitive to the number of training samples. In addition, in the real PolSAR image classification task, the huge number of training samples is not accessible. On the other hand, classification performance using just small number of labelled samples is not an accurate result. So, in this paper, aiming at the small number of training samples of the PolSAR image classification task, a novel ensemble self-supervised feature-learning (ESSFL) model is designed. The designed ESSFL can automatically extract PolSAR features conducive to PolSAR image classification with a small number of training samples. In addition, it can significantly decrease the dependence of neural network algorithms on large labelled samples of training set. First, to utilize the spatial–polarimetric features of PolSAR data perfectly, the EfficientNet-B0 is presented and utilized as the main section of the deep learning (DL) model to extract DL features of PolSAR data. Then, using an optimization function that constrains the cross-correlation matrix of various distortions of each sample to the identity matrix, the designed deep learning model can obtain the effective features of homogeneous samples gathering together and heterogeneous samples separating from each other in a self-supervised manner. Moreover, two ensemble learning (EL) models, feature-level and view-level ensemble, are proposed to increase the feature extraction capability and classification result by jointly using spatial features at different scales and polarimetric information at different bands. Finally, the stack of the obtained features and the main polarimetric information of PolSAR data can be input into classifier for the classification of PolSAR data. In this paper we use two different classifiers, the first one is the Random Forest (RF) classifier and the second one is the Support Vector Machine (SVM) classifier. The advantages of SVM include that it can be used to avoid the difficulties of using linear functions in the high-dimensional feature space, and the optimization problem is transformed into dual convex quadratic programs that are so usable for an actual image classification task. In addition, among all the available classification methods, random forests provide one of the highest accuracies. The random forest technique can also handle big data with numerous variables running into thousands. In fact, it can automatically balance data sets when a class is more infrequent than other classes in the data that is so important for a real PolSAR image classification task. Experimental results on three well-known PolSAR data sets illustrate that the designed ESSFL can extract more discriminant features using the designed deep learning model. In the end, the experiments prove that the designed ESSFL has a significant classification performance compared with different deep learning models in the case of small number of training samples, and also it can achieve a better result in the case of large number of training samples by comparison with the most of the deep learning models.

Lagrangian Duality in Convex Conic Programming with Simple Proofs

In this paper, we study Lagrangian duality aspects in convex conic programming over general convex cones. It is known that the duality in convex optimization is linked with specific theorems of alternatives. We formulate and prove the strong alternative theorems to the strict feasibility and analyze the relation between the boundedness of the optimal solution sets and the existence of the relative interior points in the feasible set. We also provide sufficient conditions under which the duality gap is zero and the optimal solution sets are unbounded. As a consequence, we obtain several new sufficient conditions that guarantee the strong duality between primal and dual convex conic programs. Our proofs are based only on fundamental convex analysis and linear algebra results.

On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems

On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems

Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg–Landau-type system in superconductivity in the absence of a magnetic field. In the first section, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points, which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section, we present some numerical results concerning the generalized method of lines applied to a Ginzburg–Landau-type equation.

Two-stage distributionally robust optimization model of integrated energy system group considering energy sharing and carbon transfer

Due to the small scale and few functions of the single integrated energy system, the absorption capacity of wind turbine and photovoltaic is limited, the ability to cope with uncertainties is weak, and the space for optimal allocation of resources is limited. To solve the coordination problem of robustness, economy, environmental protection and efficiency, this paper forms an integrated energy system group (IESG) by means of energy sharing and carbon transfer, and innovatively proposes a two-stage distributionally robust optimization model (TSDRO) based on kernel density estimation (KDE) and Wasserstein metric. Firstly, the structure of IESG with carbon capture, utilization, and storage-power-to-gas (CCUS-P2G) system is introduced. Then, the nonparametric KDE method is applied to fit the probability density functions of the forecast power error of wind turbine and photovoltaic. Wasserstein metric is used to characterize the fuzzy uncertainty set of distributions. The cumulative distribution function of KDE is taken as the center, and the obtained distance is taken as the radius to form the Wasserstein ball of probability distribution. Based on affinely adjustable policy, a correlation model of real-time variables with respect to day-ahead variables is established. Finally, according to the dual theory and convex optimization theory, the TSDRO model is reformulated into a solvable model. The simulation results show that: (1) energy sharing and carbon transfer can improve the ability of IESG to cope with uncertainty and expand the boundary of resource optimal allocation, and the minimum expected operating cost under the worst distribution is $ 40,259.94. (2) CCUS-P2G system strengthens the synergistic relationship between electricity and carbon and reduces the carbon emission of the system by 128.2 t. (3) After testing, the results obtained by nonparametric KDE are closer to the true distribution and more objective. (4) The TSDRO model is data-driven and has the advantages of high solving efficiency and low decision-making conservatism. The solution time of the TSDRO model is 73.03 % less than that of the stochastic optimization model, and the operation cost is 1.69 % less than that of the robust optimization model, which achieves the balance of economy, robustness and environmental protection of IESG.

Non-Simplicial Delaunay Meshing via Approximation by Radical Partitions

We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem is reduced to the construction of two dual convex polyhedra, inscribed and superscribed around a circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron and the sequence of the dual polyhedra converges to the inscribed polyhedron. We are interested in the case when the vertices of primal polyhedra can move or merge together, i.e., no new faces are allowed for dual polyhedra. These rules define the transformation of the set of initial spheres into the set of Delaunay spheres using radius variation and sphere movement and elimination. Existence theorems are still unavailable but we suggest a functional measuring the deviation of the convex polyhedron from the one inscribed into the paraboloid. It is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from the paraboloid. The functional's absolute minimizer is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by a simple translation. This formulation of the functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. We concentrate on the experimental confirmation of the approach viability and put aside mesh quality problems. The zero value of the gradient of the proposed functional defines a manifold describing the evolution of Delaunay spheres. Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a constraint.

ПОСТРОЕНИЕ НЕСИМПЛИЦИАЛЬНЫХ СЕТОК ДЕЛОНЕ ПОСРЕДСТВОМ АППРОКСИМАЦИИ РАДИКАЛЬНЫМИ РАЗБИЕНИЯМИ

We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem is reduced to the construction of two dual convex polyhedra, inscribed and superscribed around a circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron and the sequence of the dual polyhedra converges to the inscribed polyhedron. We are interested in the case when the vertices of primal polyhedra can move or merge together, i.e., no new faces are allowed for dual polyhedra. These rules define the transformation of the set of initial spheres into the set of Delaunay spheres using radius variation and sphere movement and elimination. Existence theorems are still unavailable but we suggest a functional measuring the deviation of the convex polyhedron from the one inscribed into the paraboloid. It is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from the paraboloid. The functional's absolute minimizer is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by a simple translation. This formulation of the functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. We concentrate on the experimental confirmation of the approach viability and put aside mesh quality problems. The zero value of the gradient of the proposed functional defines a manifold describing the evolution of Delaunay spheres. Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a constraint.

Optimal vault problem – form finding through 2D convex program

This work puts forward a form finding problem of designing a least-volume vault that is a surface structure spanning over a plane region, which via pure compression transfers a vertically tracking load to the supporting boundary. Through a duality scheme, developed recently for the design of pre-stressed membranes, the optimal vault problem is reduced to a pair of mutually dual convex problems formulated on the 2D reference region. The vault constructed upon solutions of those problems is both of minimum volume and minimum compliance; analytical examples of optimal vaults are given. Through a measure-theoretic approach, such optimal vaults are proved to solve the Prager problem of designing a 3D structure that by compression carries a transmissible load. The ground structure method applied to the convex problems furnishes a pair of discrete, conic quadratic programs leading to optimal design of grid-shells. By adopting the member-adding adaptive technique, this pair is efficiently tackled numerically. The computer method is demonstrated on a number of examples where highly precise grid-shell approximations of optimal vaults are found.

Dual semi-supervised convex nonnegative matrix factorization for data representation

Semi-supervised nonnegative matrix factorization (NMF) has received considerable attention in machine learning and data mining. A new semi-supervised NMF method, called dual semi-supervised convex nonnegative matrix factorization (DCNMF), is proposed in this paper for fully using the limited label information. Specifically, DCNMF simultaneously incorporates the pointwise and pairwise constraints of labeled samples as dual supervisory information into convex NMF, which results in a better low-dimensional data representation. Moreover, DCNMF imposes the nonnegative constraint only on the coefficient matrix but not on the base matrix. Consequently, DCNMF can process mixed-sign data, and hence enlarge the range of applications. We derive an efficient alternating iterative algorithm for DCNMF to solve the optimization, and analyze the proposed DCNMF method in terms of the convergence and computational complexity. We also discuss the relationships between DCNMF and several typical NMF based methods. Experimental results illustrate that DCNMF outperforms the related state-of-the-art NMF methods on nonnegative and mixed-sign datasets for clustering applications.

S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the 2 → 2 scattering matrix S2→2 such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider 3 + 1 dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves kℓ(s) that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves fℓ(s), for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

Joint user selection and power allocation optimization for energy-efficient MU-MIMO systems with limited feedback

The multi-user multiple-input multiple-output (MU-MIMO) systems can greatly improve the system throughput. An efficient user selection and power allocation schemes are essential to achieve the needed performance. This paper examines energy efficiency maximization under limited feedback in MU-MIMO systems. An expression for the achievable data rate is derived for RRH-user selection which is used to obtain the optimization framework for the energy efficiency (EE). The system EE is optimized through power allocation, RRH-user selection and number of antennas adjustment when the quality of service requirements and maximum transmit power constraints are satisfied. The formulated problem is NP-hard and non-convex. Based on Lagrange dual decomposition and successive convex approximation, a practical scheme is proposed to tackle the problem. It can be observed that there is an improvement of 4% in the energy efficiency of the proposed algorithm as compared with the energy efficiency of the existing algorithm. Furthermore, the computational complexity of the proposed algorithm is discussed. Finally, the proposed algorithm is validated through simulations and the results show that the proposed algorithm outperforms the existing algorithms in terms of the system EE.

Kippenhahn's Theorem for Joint Numerical Ranges and Quantum States

Kippenhahn's theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many Hermitian matrices is similarly the convex hull of a semialgebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove that the class of bases of these dual cones is closed under linear operations. The result offers a new geometric method to analyze quantum states.

Reweighted Dual Sparse Regularization and Convex Optimization for Bearing Fault Diagnosis

Performing early and accurate bearing fault diagnosis is of great significance. However, accurate extraction of the repetitive transients from noisy vibration signals is a critical issue. In this article, a reweighted dual sparse regularization (RDSR) method is proposed for bearing fault diagnosis, and the RDSR is a unified framework of the reweighted generalized minimax-concave (reGMC) penalty and the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm regularization. To be specific, the reGMC penalty and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm are both introduced in the sparse coding model. The reGMC is utilized to enhance the sparsity and overcome the underestimation deficiency of the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm while restraining the interference. Moreover, the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm is employed to simplify the model and overcome the overfitting problem, which can make the optimization process rapid and stable. Therefore, it can observably enhance the accuracy of bearing fault diagnosis and preserve the amplitude of the fault transient component. Comparisons with ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization model and the conventional spectral kurtosis (SK) show that the proposed method can preserve the spectral peak of the periodic transient components and improve the signal-to-noise ratio (SNR). Experimental case studies further verify that the proposed method can accurately estimate the periodic transient component from vibration signals, which demonstrates that the proposed method is feasible for bearing fault diagnosis.

Cost-Efficient Deployment of Storage Unit in Residential Energy Systems

With the mushrooming of distributed renewable generation, energy storage unit (ESU) is becoming increasingly important in residential energy systems. This letter proposes a fractional programming model to determine the optimal power, and energy capacities of residential ESUs. The objective function maximizes the ratio between the reduced electricity tariff, and the investment cost of ESU, ensuring the minimal payback time. A decomposition algorithm is developed to solve the fractional program based on convex optimization; the subproblem is a dual convex quadratic program which provides cutting planes, and the master problem comes down to a linear program after variable transformations. Compared to the widely used cost-minimum method, the proposed model is cost-efficient: it enjoys a higher rate of return which is usually welcomed by small consumers.

Simultaneous smoothness and simultaneous stability of a $C^\infty$ strictly convex integrand and its dual

In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three. (1) For a $C^\infty$ convex integrand $\gamma: S^n \to \mathbf{R}_+$, its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex integrand. (2) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $γ$ is stable if and only if its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is stable. (3) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $γ$ is stable. Then, for any $i$ ($0 \le i \le n$), a point $\theta_0 \in S^n$ is a non-degenerate critical point of $γ$ with Morse index $i$ if and only if its antipodal point $-\theta_0 \in S^n$ is a non-degenerate critical point of the dual convex integrand $δ$ with Morse index ($n-i$).

The O(N) S-matrix monolith

We consider the scattering matrices of massive quantum field theories with no bound states and a global O(N) symmetry in two spacetime dimensions. In particular we explore the space of two-to-two S-matrices of particles of mass m transforming in the vector representation as restricted by the general conditions of unitarity, crossing, analyticity and O(N) symmetry. We found a rich structure in that space by using convex maximization and in particular its convex dual minimization problem. At the boundary of the allowed space special geometric points such as vertices were found to correspond to integrable models. The dual convex minimization problem provides a novel and useful approach to the problem allowing, for example, to prove that generically the S-matrices so obtained saturate unitarity and, in some cases, that they are at vertices of the allowed space.