Quadratic Problem Research Articles

Computing Quadratic Eigenvalues and Solvent by a New Minimization Method and a Split-Linearization Technique

To solve quadratic eigenvalue problems (QEPs), especially the gyroscopic systems, two methods are proposed: an iterative direct detection method (DDM) of the complex eigenvalues of the original QEP, and a split-linearization method (SLM) for finding the solvent matrix, which results to a standard linear eigenvalue problem (LEP) being solved to compute all eigenvalues by the symmetry extension. Reducing the dimension to one-half, the LEP is recast in a simpler QEP involving the square of the solvent. We set up two new merit functions which are minimized to detect the complex eigenvalues from the original QEP and a simpler QEP. For each eigen-parameter the merit function consists of the Euclidean norm of each derived eigen-equation, whose vector variable is solved from a derived inhomogeneous linear system. Then, the golden section search algorithm is employed to minimize the merit functions and locate the complex eigenvalue as a local minimal point. The results are compared with that computed by the cyclic-reduction-based solvent (CRS) method.

The quadratic knapsack problem with setup

The Quadratic Knapsack Problem is a well-known generalization of the classical 0-1 knapsack problem, in which any pair of items produces a pairwise profit if both are selected. Another relevant generalization of the knapsack problem is the Knapsack Problem with Setup, in which the items are partitioned into classes, the items of a class can only be inserted into the knapsack if the corresponding class is activated, and activating a class involves a setup cost and a setup capacity reduction.Despite a rich literature on these two problems, their obvious generalization, i.e., the Quadratic Knapsack Problem with Setup, was never investigated so far. We discuss applications, mathematical models, deterministic matheuristic algorithms, and computationally evaluate their performance.

A stochastic moving ball approximation method for smooth convex constrained minimization

AbstractIn this paper, we consider constrained optimization problems with convex objective and smooth convex functional constraints. We propose a new stochastic gradient algorithm, called the Stochastic Moving Ball Approximation (SMBA) method, to solve this class of problems, where at each iteration we first take a (sub)gradient step for the objective function and then perform a projection step onto one ball approximation of a randomly chosen constraint. The computational simplicity of SMBA, which uses first-order information and considers only one constraint at a time, makes it suitable for large-scale problems with many functional constraints. We provide a convergence analysis for the SMBA algorithm using basic assumptions on the problem, that yields new convergence rates in both optimality and feasibility criteria evaluated at some average point. Our convergence proofs are novel since we need to deal properly with infeasible iterates and with quadratic upper approximations of constraints that may yield empty balls. We derive convergence rates of order $${\mathcal {O}} (k^{-1/2})$$ O ( k - 1 / 2 ) when the objective function is convex, and $${\mathcal {O}} (k^{-1})$$ O ( k - 1 ) when the objective function is strongly convex. Preliminary numerical experiments on quadratically constrained quadratic problems demonstrate the viability and performance of our method when compared to some existing state-of-the-art optimization methods and software.

Solving Max-cut Problem with a mixed penalization method for Semidefinite Programming

This paper is an attempt to exploit the opportunity of Semi Definite Programming (SDP), which is an area of convex and conic optimization. Indeed, Numerous NP-hard problems can be solveby using this approach. Hence, we intend to investigate the strength of SDP to model and provide tight relaxations of combinatorial and quadratic problems in order to present a new polynomial time algorithm for solving this robust model. This algorithm was firstly use to solve the nonlinear programs, the reason for which we search to extend it to the SDP programs. Actually, this algorithm designs the combination of two penalization methods. The first one is a primal-dual interior point (PDIM) method while the second one is a primal-dual exterior point (PDEM) method. Unlike the first method, which converges globally, the second one, also called the primal dual nonlinear rescaling method, has local super linear/quadratic convergence. Therefore, it seems appropriate to use a mixed algorithm based on the interior-exterior point method (IEPM). In fact, this resolution starts from the interior method, and at a certain level of execution, it proceeds to exterior method. Hence, a convergence evaluation function is use to know the level of permutation. Through evaluation, it has been approve that our approach is use to solve some instances of max-cut problem. This problem is a central graph theory model that occurs in many real problems and it is one of many NP-hard problems, which has attracted many researchers over the years. Then, we have used a semi definite programming solver SDPA (Semi Definite Programming Algorithm) that is modify to include the exterior point method subroutine. From the computational performance, we conclude that as the problem size increases, interior-exterior point algorithm gets relatively faster. The numerical results obtained are promising.

A decision-making model for joint energy and reserve scheduling of wind power producers with local intraday demand response exchange market

In this paper, a three-stage stochastic bi-level optimization framework is presented for optimal participation of wind power producers (WPPs) in day-ahead (DA), intraday, and balancing markets. In this framework, to leverage demand response (DR) services, a peer-to-peer (P2P) energy trading platform is implemented that allows local load aggregators (LAs) to contribute to the intraday markets improving both LAs’ and WPP’s benefits. Participating in the intraday DR exchange (IDRX) market enables WPP to purchase DR services from LAs, to reduce the penalty cost on the deviation between the day-head bidding and the real-time dispatch. A Stackelberg game for the bi-level decision-making model captures the conflict of interests between the WPP and LAs, in which, the upper level seeks to maximize WPP profit, while the lower level aims to maximize LAs’ economic surplus. The bi-level model is converted into its equivalent single-level mixed-integer quadratic problem (MIQP) employing the Karush-Kuhn-Tucker (KKT) conditions and strong duality theorem. Simulation results show that participation of the WPP in the IDRX market and employing spinning reserve and DR services for compensating the uncertainties are greatly dependent on its risk preferences and increase its expected profit in all conditions, significantly.

Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.

Stress fields at skin-stringer junctions in composite aircraft fuselages

This paper presents a semi-analytical analysis approach for the determination of stress fields in the vicinity of skin-stringer junctions in stiffened composite panels. The situation considered in this paper is representative for a typical stiffened panel in a modern composite aircraft fuselage. The analysis method employs a two-tier approach employing a global model based on CLPT on the one hand, and a local approach on the other hand in the form of a layerwise displacement formulation. This allows for the detailed computation of the stress concentrations in the vicinity of the skin-stringer junction. The layerwise formulation utilizes a discretization of the laminate layers into mathematical layers. The principle of the minimum of the total elastic potential yields the governing equations of the given problem, and an exponential approach leads to a quadratic eigenvalue problem that can be solved numerically. The analysis method shows excellent accuracy of the stress results in comparison with comparative finite element computations at a fraction of the computational time and effort that is required for numerical analyses.

An approach to mismatched disturbance rejection control for uncontrollable systems

This study focuses on the problem of optimal mismatched disturbance rejection control for uncontrollable linear discrete-time systems. In contrast to previous studies, by introducing a quadratic performance index such that the regulated state can track a reference trajectory and minimize the effects of disturbances, mismatched disturbance rejection control is transformed into a linear quadratic tracking problem. The necessary and sufficient conditions for the solvability of this problem over a finite horizon and a disturbance rejection controller are derived by solving a forward–backward difference equation. In the case of an infinite horizon, a sufficient condition for the stabilization of the system is obtained under the detectable condition. Additionally, in combination with the generalized extended state observer, a controller design method is proposed, and the stability analysis of the system under this controller is presented. This paper details our novel approach to disturbance rejection. Finally, four examples are provided to demonstrate the effectiveness of the proposed method.

A novel successive convexification framework with symplectic pseudospectral solutions for nonlinear optimal control problems of constrained pantograph delay systems

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses.

A Stable Solution of a Nonuniformly Perturbed Quadratic Minimization Problem by the Extragradient Method with Step Size Separated from Zero

A Stable Solution of a Nonuniformly Perturbed Quadratic Minimization Problem by the Extragradient Method with Step Size Separated from Zero

Stochastic linear–quadratic control problems with affine constraints

This paper investigates the stochastic linear–quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality theory, both the dual problem and the state feedback form of the solution are obtained for the primal problem. Under the Slater condition, the strong duality is proved between the dual problem and the primal problem, and the KKT condition is also provided for solving the primal problem. Moreover, a new sufficient condition is given for the invertibility assumption, which ensures the uniqueness of the solutions to the dual problem. Finally, two numerical examples are provided to illustrate our main results.

OPT-FRAC-CHN: Optimal Fractional Continuous Hopfield Network

The continuous Hopfield network (CHN) is a common recurrent neural network. The CHN tool can be used to solve a number of ranking and optimization problems, where the equilibrium states of the ordinary differential equation (ODE) related to the CHN give the solution to any given problem. Because of the non-local characteristic of the “infinite memory” effect, fractional-order (FO) systems have been proved to describe more accurately the behavior of real dynamical systems, compared to the model’s ODE. In this paper, a fractional-order variant of a Hopfield neural network is introduced to solve a Quadratic Knap Sac Problem (QKSP), namely the fractional CHN (FRAC-CHN). Firstly, the system is integrated with the quadratic method for fractional-order equations whose trajectories have shown erratic paths and jumps to other basin attractions. To avoid these drawbacks, a new algorithm for obtaining an equilibrium point for a CHN is introduced in this paper, namely the optimal fractional CHN (OPT-FRAC-CHN). This is a variable time-step method that converges to a good local minima in just a few iterations. Compared with the non-variable time-stepping CHN method, the optimal time-stepping CHN method (OPT-CHN) and the FRAC-CHN method, the OPT-FRAC-CHN method, produce the best local minima for random CHN instances and for the optimal feeding problem.

A new Fermatean fuzzy multi‐objective indefinite quadratic transportation problem with an application to sustainable transportation

AbstractIn this work, a Fermatean fuzzy (FF) multi‐objective indefinite quadratic transportation problem (TP) is introduced. Due to some unavoidable reasons, real‐life transportation parameters such as supply, demand and costs are indeterminate in nature and cannot be expressed in crisp terms. We represent these parameters using FF numbers, an extension of fuzzy numbers, which are capable of representing indeterminacy efficiently. A multi‐objective indefinite quadratic TP where each objective is a product of two linear factors (cost functions) is considered. Defuzzification of FF numbers is accomplished by the introduction of ‐cut for the first time. The obtained crisp TP is solved using the intuitionistic fuzzy programming approach and FF programming approach to arrive at a compromise solution. To substantiate the work, solution methodology based on defuzzification using the ranking function is also deliberated. The applicability of the model is demonstrated through a sustainable TP, which simultaneously minimizes transportation cost with depreciation cost and packaging cost with wastage cost. The resulting value of the objective functions and the aspiration levels are compared to depict the efficacy of the proposed method over the ranking function method. The concluding section summarizes the work, and future avenues along with some limitations of the work are also specified.

Plug-and-play analytical paradigm for the scattering of plane waves by ‘layer-cake’ periodic systems

We investigate the scattering of scalar plane waves in two dimensions by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic laminae, we develop a solution of the scattering problem by merging the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each lamina. The featured Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEVP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary—that is preserved throughout the system. Spectral analysis of the QEVP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate that the contribution of individual eigenvalues (and so eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch (FB)-wave solution. The proposed methodology caters for the optimal design of rainbow traps, energy harvesters and metasurfaces, whose potency to manipulate waves is decided not only by the individual dispersion characteristics of the component laminae, but also by ordering and generally fitting of the latter into a composite layer. Using the FB approach, evaluation of trial configurations—as generated by the permutation and window translation/stretching of the component laminae—can be accelerated by decades.

Linear–quadratic stochastic Volterra controls I: Causal feedback strategies

In this paper, we formulate and investigate the notion of causal feedback strategies arising in linear–quadratic control problems for stochastic Volterra integral equations (SVIEs) with singular and non-convolution-type coefficients. We show that there exists a unique solution, which we call the causal feedback solution, to the closed-loop system of a controlled SVIE associated with a causal feedback strategy. Furthermore, introducing two novel equations named a Type-II extended backward stochastic Volterra integral equation and a Lyapunov–Volterra equation, we prove a duality principle and a representation formula for a quadratic functional of controlled SVIEs in the framework of causal feedback strategies.

Phase response constrained symbol‐level waveform design for dual‐functional radar‐communication systems

AbstractIn this letter, a novel algorithm is proposed to design symbol‐level waveform for dual‐functional radar‐communication (DFRC) systems with low range sidelobe. Different from current schemes design waveform at the block level, a phase response constraint is introduced at the symbol level to provide an additional degree of freedom to decrease the range sidelobe in the pulse compression procedure, which is highly desired in radar systems. In particular, the phase response at the target direction is constrained to be similar to the given reference phase, and the matching error between the designed beampattern and the desired one is minimized subject to the constant modulus constraint. Furthermore, to guarantee the quality of service for communication, constructive interference is exploited at the symbol level for each communication user. An alternating direction method of multipliers algorithm is also proposed to solve the resulting nonconvex optimization problem with tractable subproblems solved sufficiently by the manifold optimization and standard quadratic problem. Numerical simulation results demonstrate the performance of the proposed method.

Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems

We start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear–quadratic (LQ) stochastic control problem. Using the well-developed theory on LQ problems, we describe the solution to the obtained LQ one and translate it back to the solution for the (extended) initial trade execution problem. Finally, we illustrate our results by several examples. Among other things, the examples discuss the Obizhaeva–Wang model with random (terminal and moving) targets, the necessity to extend the initial trade execution problem to a reasonably large class of progressively measurable processes (even going beyond semimartingales), and the effects of diffusive components in the price impact process and/or the resilience process.

An accelerated lyapunov function for Polyak’s Heavy-ball on convex quadratics

AbstractIn 1964, Polyak showed that the Heavy-ball method, the simplest momentum technique, accelerates convergence of strongly-convex problems in the vicinity of the solution. While Nesterov later developed a globally accelerated version, Polyak’s original algorithm remains simpler and more widely used in applications such as deep learning. Despite this popularity, the question of whether Heavy-ball is also globally accelerated or not has not been fully answered yet, and no convincing counterexample has been provided. This is largely due to the difficulty in finding an effective Lyapunov function: indeed, most proofs of Heavy-ball acceleration in the strongly-convex quadratic setting rely on eigenvalue arguments. Our work adopts a different approach: studying momentum through the lens of quadratic invariants of simple harmonic oscillators. By utilizing the modified Hamiltonian of Störmer-Verlet integrators, we are able to construct a Lyapunov function that demonstrates an $$O(1/k^2)$$ O ( 1 / k 2 ) rate for Heavy-ball in the case of convex quadratic problems. Our novel proof technique, though restricted to linear regression, is found to work well empirically also on non-quadratic convex problems, and thus provides insights on the structure of Lyapunov functions to be used in the general convex case. As such, our paper makes a promising first step towards potentially proving the acceleration of Polyak’s momentum method and we hope it inspires further research around this question.

Accelerate rotation invariant sliced Gromov-Wasserstein distance by an alternative optimization method

The traditional Wasserstein distance is inadequate for comparing two distributions in different metric spaces. The Gromov-Wasserstein distance (GWD), which evolves from the Gromov-Hausdorff distance, constructs intraspace pairs to ensure comparability. However, solving the GWD requires tackling a complex nonconvex quadratic problem, a process that typically demands significant time and space complexity. The sliced Gromov-Wasserstein distance (SGW) addresses the efficiency issues of GWD by employing a projection-slice method. Unfortunately, slicing destroys the crucial rotation invariance property of GWD. The rotation invariant sliced Gromov-Wasserstein distance (RISGW) restores it by enumerating the alignments of the distributions. Inspired by our observations, we introduce BRISGW, a novel batch rotation invariant sliced Gromov-Wasserstein distance that alternately iterates and approximates the underlying best alignment both effectively and efficiently. We theoretically prove the feasibility the proposed our approach. Moreover, significant experimental results show that the proposed method outperforms the state-of-the-art methods.

A random active set method for strictly convex quadratic problem with simple bounds

The active set method aims at finding the correct active set of the optimal solution and it is a powerful method for solving strictly convex quadratic problems with bound constraints. To guarantee the finite step convergence, existing active set methods all need strict conditions or some additional strategies, which can significantly impact the efficiency of the algorithm. In this paper, we propose a random active set method that introduces randomness in the active set’s update process. We prove that the algorithm can converge in a finite number of iterations with probability one, without any extra conditions on the problem or any supplementary strategies. At last, numerical experiments show that the algorithm can obtain the correct active set within a few iterations, and it has better efficiency and robustness than the existing methods.