Reformulation Of Problem Research Articles

Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation

The mean-variance (MV) portfolio is typically formulated as a quadratic programming (QP) problem that linearly combines the conflicting objectives of minimizing the risk and maximizing the expected return through a risk aversion profile parameter. In this formulation, the two objectives are expressed in different units, an issue that could definitely hamper obtaining a more competitive set of portfolio weights. For example, a modification in the scale in which returns are expressed (by one or percent) in the MV portfolio, implies a modification in the solution of the problem. Motivated by this issue, a novel mean squared variance (MSV) portfolio is proposed in this paper. The associated optimization problem of the proposed strategy is very similar to the Markowitz optimization, with the exception of the portfolio mean, which is presented in squared form in our formulation. The resulting portfolio model is a non-convex QP problem, which has been reformulated as a mixed-integer linear programming (MILP) problem. The reformulation of the initial non-convex QP problem into an MILP allows for future researchers and practitioners to obtain the global solution of the problem via the use of current state-of-the-art MILP solvers. Additionally, a novel purely data-driven method for determining the optimal value of the hyper-parameter that is associated with the MV and MSV approaches is also proposed in this paper. The MSV portfolio has been empirically tested on eight portfolio time series problems with three different estimation windows (composing a total of 24 datasets), showing very competitive performance in most of the problems.

Analysis of a stabilized finite element approximation for a linearized logarithmic reformulation of the viscoelastic flow problem

In this paper we present the numerical analysis of a finite element method for a linearized viscoelastic flow problem. In particular, we analyze a linearization of the logarithmic reformulation of the problem, which in particular should be able to produce results for Weissenberg numbers higher than the standard one. In order to be able to use the same interpolation for all the unknowns (velocity, pressure and logarithm of the conformation tensor), we employ a stabilized finite element formulation based on the Variational Multi-Scale concept. The study of the linearized problem already serves to show why the logarithmic reformulation performs better than the standard one for high Weissenberg numbers; this is reflected in the stability and error estimates that we provide in this paper.

Selective Image Segmentation Models Using Three Distance Functions

Image segmentation can be defined as partitioning an image that contains multiple segments of meaningful parts for further processing. Global segmentation is concerned with segmenting the whole object of an observed image. Meanwhile, the selective segmentation model is concerned with segmenting a specific object required to be extracted. The Convex Distance Selective Segmentation (CDSS) model, which uses the Euclidean distance function as the fitting term, was proposed in 2015. However, the Euclidean distance function takes time to compute. This paper proposes the reformulation of the CDSS minimization problem by changing the fitting term with three popular distance functions, namely Chessboard, City Block, and Quasi-Euclidean. The proposed models are CDSSNEW1, CDSSNEW2, and CDSSNEW3, which apply the Chessboard, City Block, and Quasi-Euclidean distance functions respectively. In this study, the Euler-Lagrange (EL) equations of the proposed models were derived and solved using the Additive Operator Splitting method. Then, MATLAB coding was developed to implement the proposed models. The accuracy of the segmented image was evaluated using the Jaccard (JSC) and Dice Similarity Coefficients (DSC). The execution time was recorded to measure the efficiency of the models. Numerical results showed that the proposed CDSSNEW1 model based on the Chessboard distance function could segment the specific object successfully for all grayscale images with the fastest execution time compared to other models.

Reformulation of transmission and reflection problems in terms of quantum wave impedance function

Reformulation of transmission and reflection problems in terms of quantum wave impedance function

Optimal Risk Operation for a Coupled Electricity and Heat System Considering Different Operation Modes

The coupled electricity and heat system (CEHS) is considered one of the most efficient energy utilization schemes for the energy transition to solve energy crises and environmental problems. However, the individual data between the power system (PS) and heat system (HS) probably limit the high share and optimization operation. Moreover, the uncertain renewables and complex coupled network create challenges regarding the operation risk of CEHS. Given that the CEHS may be affiliated with different operation entities, this paper proposes two operation modes to achieve the solution of the optimal risk operation model (OROM), including the distributionally robust chance constraint (DRCC)-based centralized risk operation mode (CROM) and the DRCC-based alternating direction method of multipliers (ADMM) distributed risk operation mode (DROM). By formulating the moment-based ambiguity set estimated from historical data and introducing the operation risk constraints, a tractable reformulation of two-stage DRCC OROM problems is presented under CROM. Moreover, the DRCC-based ADMM distributed algorithm with the guarantee of convergence is developed to optimize the PS and HS independently under DROM considering the operation risk. The two operation modes achieve the minimum amount of information shared between the two systems regarding the underlying distribution. In the numerical case, the approximate OROM solution is obtained between CROM and DROM. The two proposed approaches based on different operation modes outperform the existing methods according to the risk level depicted by different forms of the violation probability and the risk cost compared with Gaussian chance constraint (GCC) under CROM, while the safety coefficient $\epsilon $ is set as 0.25. Then, the impact of the iteration number on the convergence is also discussed with comparison of the classical ADMM under DROM.

Quadratically adjustable robust linear optimization with inexact data via generalized S-lemma: Exact second-order cone program reformulations

• Studies adjustable robust linear optimization problems with quadratic decision rules. • Incorporates inexactness of the data in the construction of uncertainty sets. • Establishes generalizations of S-Lemma for classes of nonconvex quadratic systems. • Presents exact conic programming reformulations via a generalized S-Lemma. • Illustrates results with numerical experiments on lot-sizing problems with demand uncertainty. Adjustable robust optimization allows for some variables to depend upon the uncertain data after its realization. However, the uncertainty is often not revealed exactly. Incorporating inexactness of the revealed data in the construction of ellipsoidal uncertainty sets, we present an exact second-order cone program reformulation for robust linear optimization problems with inexact data and quadratically adjustable variables. This is achieved by establishing a generalization of the celebrated S-lemma for a separable quadratic inequality system with at most one non-homogeneous function. It allows us to reformulate the resulting separable quadratic constraints over an intersection of two ellipsoids in terms of second-order cone constraints. We illustrate our results via numerical experiments on adjustable robust lot-sizing problems with demand uncertainty, showing improvements over corresponding problems with affinely adjustable variables as well as with exactly revealed data.

Robust structured H2 synthesis for linear systems subject to time-invariant uncertainties with global optimization

Abstract This paper proposes a new way to solve the structured H2 synthesis problem under time invariant plant uncertainties. A worst-case minimization approach enables to formulate the synthesis problem as a min-max problem subject to quantified constraints. We propose a global optimization algorithm based on Interval Analysis to solve this problem. This algorithm provides an upper and a lower bound on a worst-case H2 norm objective function. The originality of the paper relies on the reformulation of the optimization problem as a min-max problem, providing bounds on the worst-case H2 norm.

A Unified SO(3) Approach to the Attitude Control Design for Quadrotors

This paper proposes a two-phased problem reformulation approach to solve the quadrotor UAV attitude control problem. Given an initial attitude, the objective is to design a control to drive it toward a desired value. The first phase begins with the definition of three types of errors pertaining to attitude discrepancy, followed by the utilization of virtual control with which the dynamics of quadrotor’s attitude and angular velocity are then unified under the framework of SO(3), rendering the original control problem much more transparent to tackle. Through cancellation of certain unwanted terms emerging from the unification process, entry-wise treatment of the remaining matrix dynamics is then employed in the second phase to transform it further into the stabilization problem of a 3-dimensional linear time-invariant system that is fairly easy to solve since to which standard feedback designs such as linear quadratic regulator and pole placement are readily applicable. In addition to the classical designs, a specialized and structurally simple controller is provided to reduce the number of gains. To account for system parametric uncertainties, external disturbances, and sensor measurement errors and noises, a second controller based on classical $H_{\infty }$ theory is presented to enhance the robustness of the design. A numerical example is given, and computer simulations are conducted to generate error and control trajectories to assess and compare the effectiveness of the presented designs.

Geometric Convergence of Gradient Play Algorithms for Distributed Nash Equilibrium Seeking

We study distributed algorithms for seeking a Nash equilibrium in a class of convex networked Nash games with strongly monotone mappings. Each player has access to her own smooth local cost function and can communicate to her neighbors in some undirected graph. To deal with fast distributed learning of Nash equilibria under such settings, we introduce a so called augmented game mapping and provide conditions under which this mapping is strongly monotone. We consider a distributed gradient play algorithm for determining a Nash equilibrium (GRANE). The algorithm involves every player performing a gradient step to minimize her own cost function while sharing and retrieving information locally among her neighbors in the network. Using the reformulation of the Nash equilibrium problem based on the strong monotone augmented game mapping, we prove the convergence of this algorithm to a Nash equilibrium with a geometric rate. Furthermore, we introduce the Nesterov type acceleration for the gradient play algorithm. We demonstrate that, similarly to the accelerated algorithms in centralized optimization and variational inequality problems, our accelerated algorithm outperforms GRANE in the convergence rate. Moreover, to relax assumptions required to guarantee the strongly monotone augmented mapping, we analyze the restricted strongly monotone property of this mapping and prove geometric convergence of the distributed gradient play under milder assumptions.

Extended group finite element method

Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation interpolates nonlinear terms onto the finite element approximation space, we propose the use of a separate approximation space that is tailored to the nonlinearity. In many cases, this allows for the exact reformulation of the discrete nonlinear problem into a quadratic problem with algebraic constraints. Furthermore, the substitution of the nonlinear terms often shifts general nonlinear forms into trilinear forms, which can easily be described by third-order tensors. The numerical benefits as well as the advantages in comparison to the original group finite element method are studied using a wide variety of academic benchmark problems.

Analytic Deep Learning-Based Surrogate Model for Operational Planning With Dynamic TTC Constraints

The increased penetration of wind power introduces more operational changes of critical corridors and the traditional time-consuming transient stability constrained total transfer capability (TTC) operational planning is unable to meet the real-time monitoring need. This paper develops a more computationally efficient approach to address that challenge via the analytical deep learning-based surrogate model. The key idea is to resort to deep learning for developing a computationally cheap surrogate model to replace the original time-consuming differential-algebraic constraints related to TTC. However, the deep learning-based surrogate model introduces implicit rules that are difficult to handle in the optimization process. To this end, we derive the Jacobian and Hessian matrices of the implicit surrogate models and finally transfer them into an analytical formulation that can be easily solved by the interior point method. Surrogate modeling and problem reformulation allow us to achieve significantly improved computational efficiency and the yielded solutions can be used for operational planning. Numerical results carried out on the modified IEEE 39-bus and 68-bus systems demonstrate the effectiveness of the proposed method in dealing with complicated TTC constraints while balancing the computational efficiency and accuracy.

Hybrid GA-SOCP Approach for Placement and Sizing of Distributed Generators in DC Networks

This research addresses the problem of the optimal location and sizing distributed generators (DGs) in direct current (DC) distribution networks from the combinatorial optimization. It is proposed a master–slave optimization approach in order to solve the problems of placement and location of DGs, respectively. The master stage applies to the classical Chu & Beasley genetic algorithm (GA), while the slave stage resolves a second-order cone programming reformulation of the optimal power flow problem for DC grids. This master–slave approach generates a hybrid optimization approach, named GA-SOCP. The main advantage of optimal dimensioning of DGs via SOCP is that this method makes part of the exact mathematical optimization that guarantees the possibility of finding the global optimal solution due to the solution space’s convex structure, which is a clear improvement regarding classical metaheuristic optimization methodologies. Numerical comparisons with hybrid and exact optimization approaches reported in the literature demonstrate the proposed hybrid GA-SOCP approach’s effectiveness and robustness to achieve the global optimal solution. Two test feeders compose of 21 and 69 nodes that can locate three distributed generators are considered. All of the computational validations have been carried out in the MATLAB software and the CVX tool for convex optimization.

Time-periodic solutions of symmetric hyperbolic systems

We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, provided that all the characteristics with respect to the direction normal to the planes have the same sign. It is also shown that global-in-time solutions to initial-boundary value problems coincide with the solutions to corresponding time-periodic problems after a finite time. We devote one section to the reformulation of several realistic problems and see our results have wide applicability.

On recent progress of single-realization recoveries of random Schrödinger systems

<p style='text-indent:20px;'>We consider the recovery of some statistical quantities by using the near-field or far-field data in quantum scattering generated under a single realization of the randomness. We survey the recent main progress in the literature and point out the similarity among the existing results. The methodologies in the reformulation of the forward problems are also investigated. We consider two separate cases of using the near-field and far-field data, and discuss the key ideas of obtaining some crucial asymptotic estimates. We pay special attention on the use of the theory of pseudodifferential operators and microlocal analysis needed in the proofs.</p>

Distributed Algorithm for Base Station Assignment in 4G/5G Machine-Type Communication Scenarios with Backhaul Limited Conditions.

A progressive paradigm shift from centralized to distributed network architectures has been consolidated since the 4G communication standard, calling for novel decision-making mechanisms with distributed control to operate at the network edge. This situation implies that each base station (BS) must manage resources independently to meet the quality of service (QoS) of existing human-type communication devices (HTC), as well as the emerging machine type communication (MTC) devices from the internet of things (IoT). In this paper, we address the BS assignment problem, whose aim is to determine the most appropriate serving BS to each mobile device. This problem is formulated as an optimization problem for maximizing the system throughput and imposing constraints on the air interface and backhaul resources. The assignment problem is challenging to solve, so we present a simple yet valid reformulation of the original problem while using dual decomposition theory. Subsequently, we propose a distributed price-based BS assignment algorithm that performs at each BS the assignment process, where a novel pricing update scheme is presented. The simulation results show that our proposed solution outperforms traditional maximum signal to interference plus noise ratio (Max-SINR) and minimum path-loss (Min-PL) approaches in terms of system throughput.

Voltage Stability Analysis in Medium-Voltage Distribution Networks Using a Second-Order Cone Approximation

This paper addresses the voltage stability margin calculation in medium-voltage distribution networks in the context of exact mathematical modeling. This margin calculation is performed with a second-order cone (SOCP) reformulation of the classical nonlinear non-convex optimal power flow problems. The main idea around the SOCP approximation is to guarantee the global optimal solution via convex optimization, considering as the objective function the λ-coefficient associated with the maximum possible increment of the load consumption at all the nodes. Different simulation cases are considered in one test feeder, described as follows: (i) the distribution network without penetration of distributed generation; (ii) the distribution network with penetration of distributed generation; and (iii) the distribution grid with capacitive compensation. Numerical results in the test system demonstrated the effectiveness of the proposed SOCP approximation to determine the λ-coefficient. In addition, the proposed approximation is compared with nonlinear tools available in the literature. All the simulations are carried out in the MATLAB software with the CVX package and the Gurobi solver.

Alignment of biological networks by integer linear programming: virus-host protein-protein interaction networks

BackgroundThe alignment of protein-protein interaction networks was recently formulated as an integer quadratic programming problem, along with a linearization that can be solved by integer linear programming software tools. However, the resulting integer linear program has a huge number of variables and constraints, rendering it of no practical use.ResultsWe present a compact integer linear programming reformulation of the protein-protein interaction network alignment problem, which can be solved using state-of-the-art mathematical modeling and integer linear programming software tools, along with empirical results showing that small biological networks, such as virus-host protein-protein interaction networks, can be aligned in a reasonable amount of time on a personal computer and the resulting alignments are structurally coherent and biologically meaningful.ConclusionsThe implementation of the integer linear programming reformulation using current mathematical modeling and integer linear programming software tools provided biologically meaningful alignments of virus-host protein-protein interaction networks.

Robust fault tolerant optimal predictive control of hybrid actuators with time‐varying delay for industrial robot arm

AbstractThis paper presents a robust hybrid fault‐tolerant optimal predictive control (HFTPC), for an industrial robot arm under hybrid (electric and pneumatic) actuator faults and/or varying time‐delays. Based on the error dynamics, estimated states, and a predictive controller, a new state feedback control law is proposed and implemented for the reformulation of the optimal control problem of a nonlinear faulty hybrid actuator system based on predictive control via linear matrix inequalities (LMIs). First, a robust MPC scheme is performed in which the future control sequence is used to compensate the varying time‐delays. Then, a robust stable hybrid fault tolerant predivtive control is implemented to handle actuators faults to effect robust trajectory tracking. In fact, the stability of hybrid systems based on the proposed control scheme is a very sensitive criterion. Therefore, stability conditions are required for controlling the industrial arm under faulty hybrid (electric and pneumatic) actuator, based on the Lyapunov‐Krasovskii (L‐K) theory, less conservative stable conditions in terms of LMIs are given and used to ensure the asymptotically robust stability of closed‐loop constrained system that dependent delay‐range. To highlight the robustness and effectiveness of the proposed approach, a simulation study of an industrial robot arm example is proved, where the results showed the prompt and the accuracy of the proposed scheme.

Proactive optimization of transmission power and 3D trajectory in UAV-assisted relay systems with mobile ground users

In this paper, an Unmanned Aerial Vehicle (UAV)-assisted relay communication system is studied, where a UAV is served as a flying relay to maintain a communication link between a mobile source node and a remote destination node. Specifically, an average outage probability minimization problem is formulated firstly, with the constraints on the transmission power of the source node, the maximum energy consumption budget, the transmission power, the speed and acceleration of the flying UAV relay. Next, the closed-form of outage probability is derived, under the hybrid line-of-sight and non-line-of-sight probability channel model. To deal with the formulated nonconvex optimization, a long-term proactive optimization mechanism is developed. In particular, firstly, an approximation for line-of-sight probability and a reformulation of the primal problem are given, respectively. Then, the reformulated problem is transformed into two subproblems: one is the transmission power optimization with given UAV’s trajectory and the other is the trajectory optimization with given transmission power allocation. Next, two subproblems are tackled via tailoring primal–dual subgradient method and successive convex approximation, respectively. Furthermore, a proactive optimization algorithm is proposed to jointly optimize the transmission power allocation and the three-dimensional trajectory. Finally, simulation results demonstrate the performance of the proposed algorithm under various parameter configurations.

Joint Target Assignment and Resource Optimization Framework for Multitarget Tracking in Phased Array Radar Network

In this article, a joint target assignment and resource optimization (JTARO) strategy is proposed for the application of multitarget tracking in phased array radar network system. The key mechanism of our proposed JTARO strategy is to employ the optimization technique to jointly optimize the target-to-radar assignment, revisit time control, bandwidth, and dwell time allocation subject to several resource constraints, while achieving better tracking accuracies of multiple targets and low probability of intercept (LPI) performance of phased array radar network. The analytical expression for Bayesian Cramér-Rao lower bound with the aforementioned adaptable parameters is calculated and subsequently adopted as the performance metric for multitarget tracking. After problem partition and reformulation, an efficient three-stage solution methodology is developed to resolve the underlying mixed-integer, nonlinear, and nonconvex optimization problem. To be specific, in Step 1, the revisit time for each target is determined. In Step 2, we implement the joint signal bandwidth and dwell time allocation for fixed target-to-radar assignments, which combine the cyclic minimization algorithm and interior point method. In Step 3, the optimal target-to-radar assignments are obtained, which results in the minimization of both the tracking accuracy for multiple targets and the total dwell time consumption of the network system. Simulation results are provided to demonstrate the advantages of the presented JTARO strategy, in terms of the achievable multitarget tracking accuracy and LPI performance of phased array radar network.