Parametric Quadratic Programming Research Articles

Fast quadratic model predictive control based on sensitivity analysis and Wolfe method

AbstractThis paper proposes a new algorithm based on sensitivity analysis and the Wolfe method to solve a sequence of parametric quadratic programming (QP) problems such as those that arise in quadratic model predictive control (QMPC). The Wolfe method, based on Karush–Kuhn–Tucker conditions, has been used to convert parametric QP problems to parametric linear programming (LP) problems, and then the sensitivity analysis is applied to solve the sequence of the parametric LP problems. This strategy obtains sensitivity analysis‐based QMPC (SA‐QMPC) algorithm. It is proved that the computational complexity of SA‐QMPC is for a region of the initial conditions and for sufficiently small sampling time and all initial conditions, where and are the horizon time and dimension of the state vector, respectively. Numerical results indicate the potential and properties of the proposed algorithm.

DIRECTIONAL DIFFERENTIABILITY OF THE OPTIMAL VALUE IN QUADRATIC PROGRAMMING PROBLEMS UNDER LINEAR CONSTRAINTS ON HILBERT SPACES

We investigate the first-order directional differentiability of the optimal value function in parametric quadratic programming problems under linear constraints in Hilbert spaces. We derive an explicit formula for computing the directional derivative of the optimal value function in cases where the quadratic part of the objective function is in Legendre form.

Stability Verification of Neural Network Controllers Using Mixed-Integer Programming

We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a Mixed-Integer Quadratic Program (MIQP). Additionally, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU Neural Networks (NNs), optimal solution maps of parametric quadratic programs, and Model Predictive Control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a DC-DC power converter case study and investigate its computational complexity.

Mean–variance value at risk criterion for solving a p-median location problem on networks with type-2 intuitionistic fuzzy weights

In this paper, we consider the p-median location problem on networks in which the vertex weights are the type-2 intuitionistic fuzzy variables. First, the credibilistic value at risk of the intuitionistic fuzzy variables is introduced. Using value at risk criterion, we define a new method for reducing of type-2 intuitionistic fuzzy variables to intuitionistic fuzzy variables that is named the value at risk reduced intuitionistic fuzzy variables. Further, the mean and variance values of value at risk reduced intuitionistic fuzzy variables are described. Finally, we apply the mean–variance criterion for solving the type-2 intuitionistic fuzzy p-median location problem. Also, it is shown that this problem is equivalent to a parametric quadratic convex programming problem.

Theoretically Scrutinizing Kinks on Efficient Frontiers and Computationally Reporting Nonexistence of the Tangent Portfolio for the Capital Asset Pricing Model by Parametric-Quadratic Programming

Researchers traditionally compute isolated points for an efficient frontier and assume a line which passes through a risk-free asset [Formula: see text] and is tangent to the frontier. The tangency plays pivotal roles for the capital asset pricing model (CAPM). However, the assumption may not hold in the presence of kinks (as non-differentiable points) on efficient frontiers. Kinks are detected by parametric-quadratic programming only and not by ordinary portfolio optimization. Up until now, there has been no research to theoretically scrutinize kink properties (especially implications to CAPM) and systematically quantify the nonexistence of the tangency. In such an area, this paper contributes to the literature. In theorems and corollaries, we prove the nonexistence of the tangency and substantiate that expected-return axis is composed of piecewisely connected intervals for which the tangency does not exist and intervals for which the tangency exists. Computationally, we reveal universal existence of kinks (e.g., 0.2 to 8.0 kinks for 5-stock to 1800-stock portfolio selections) and the tangency-nonexistence ratios as about 0.066.

Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path” where the discontinuous ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-function provides the exact sparsity count; the “ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function provides a convex surrogate of sparsity count; and the “capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the nonconvex nondifferentiable capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function aims to enhance the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Our study of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-regularized problem offers interesting theoretical properties and practical compromise between the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path and the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Indeed, while the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function.

Robust Markowitz: Comprehensively maximizing Sharpe ratio by parametric-quadratic programming

<p style='text-indent:20px;'>Markowitz formulates portfolio selection and calls the optimal solutions as an efficient frontier. Sharpe initiates Sharpe ratio for frontier portfolios' reward to variability. Finance textbooks assume that there exists a line which passes through a risk-free rate and is tangent to an efficient frontier. The tangent portfolio enjoys the maximum Sharpe ratio.</p><p style='text-indent:20px;'>However, the assumption is over-simplistic because we prove that other situations exist. For example, Sharpe ratio itself may not be even well-defined. We comprehensively maximize Sharpe ratio. In such an area, this paper contributes to the literature. Specifically, we identify the other situations by parametric-quadratic programming which renders complete efficient frontiers by piecewise-hyperbola structure. Researchers traditionally view efficient frontiers by just isolated points. We accomplish handy formulae, so investors can even manually process them.</p><p style='text-indent:20px;'>The COVID-19 pandemic is unleashing crises. Unfortunately, there is quite limited research of portfolio selection for COVID. In such an area, this paper contributes to the practice. Specifically, we originate a counter-COVID measure for stocks and integrate it as a constraint into portfolio-selection models. The maximum-Sharpe-ratio portfolio outperforms stock-market indexes in sample. We launch the models for Dow Jones Industrial Average and discover outperformance out of sample.</p>

A kernel path algorithm for general parametric quadratic programming problem

It is well known that the performance of a kernel method highly depends on the choice of kernel parameter. A kernel path provides a compact representation of all optimal solutions, which can be used to choose the optimal value of kernel parameter along with cross validation (CV) method. However, none of these existing kernel path algorithms provides a unified implementation to various learning problems. To fill this gap, in this paper, we first study a general parametric quadratic programming (PQP) problem that can be instantiated to an extensive number of learning problems. Then we provide a generalized kernel path (GKP) for the general PQP problem. Furthermore, we analyze the iteration complexity and computational complexity of GKP. Extensive experimental results on various benchmark datasets not only confirm the identity of GKP with several existing kernel path algorithms, but also show that our GKP is superior to the existing kernel path algorithms in terms of generalization and robustness.

SDO and LDO relaxation approaches to complex fractional quadratic optimization

This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known classical Dinkelbach method. Then a semidefinite and Lagrangian dual optimization approaches are presented to solve the nonconvex parametric problem at each iteration of the bisection and generalized Newton algorithms. Finally, the numerical results demonstrate the effectiveness of the proposed approaches.

Parallel MPC for Linear Systems With Input Constraints

This article is about a real-time model predictive control algorithm for large-scale, structured linear systems with polytopic control constraints. The proposed controller receives the current state measurement as an input, and computes a suboptimal control reaction by evaluating a finite number of piecewise affine functions that correspond to the explicit solution maps of small-scale parametric quadratic programming (QP) problems. We provide asymptotic stability guarantees, which can be verified offline. The feedback controller is computing approximations of the optimal input, because we are enforcing real-time requirements assuming that it is not possible to solve the given large-scale QP in the given amount of time. Here, a key contribution of this article is that we provide a bound on the suboptimality of the controller. The approach is illustrated by benchmark case studies.

Optimal robust fault detection of discrete‐time LPV systems with measurement error‐affected scheduling variables combining ZKF and pQP

SummaryOptimal robust state estimation (SE) and fault detection (FD) methods of discrete‐time linear parameter varying systems with measurement error‐affected scheduling variables are proposed under the boundedness assumption of system uncertainties. By using the weighted Frobenius norm of the generator matrix of SE zonotope to characterize the set size, the optimal observer gain can be computed by a zonotopic Kalman filter (ZKF) procedure for the purpose of observation. Meanwhile, by minimizing the influence of system uncertainties while maximizing that of faults on SE to enhance the sensitivity of FD, the optimal FD criterion is characterized based on an on‐line fractional programming problem, which can be equivalently transformed into a parametric quadratic programming (pQP) problem. The pQP problem can be efficiently solved by searching the root of its nonlinear characteristic equation using secant method. In general, as long as sensors with sufficiently high precision are equipped to measure the scheduling variables, the bounds of measurement errors of scheduling variables can be less conservative than those direct bounds of scheduling variables, which can reduce the conservatism of FD or SE in this way. At the end of this article, a case study based on a practical circuit model is used to illustrate the effectiveness of the proposed method.

Approximate Closed-Loop Robust Model Predictive Control With Guaranteed Stability and Constraint Satisfaction

The real-time implementation of closed-loop robust model predictive control (MPC) schemes is an important challenge for fast systems, as their solution complexity depends strongly on the system size, control policy parametrization, and prediction horizon. We look to address this problem by approximating the implicitly-defined MPC controller using deep learning. Although the resulting neural network approximation has a small memory footprint and can be efficiently computed, it does not guarantee robust constraint satisfaction or stability. We propose a novel projection-based strategy that is capable of providing a certificate of robust feasibility and input-to-state stability in real-time. We also show how this projection operator can be formulated as a parametric quadratic program that is solvable offline. The advantages of the proposed approach are demonstrated on a benchmark case study.

A new quadratic deviation of fuzzy random variable and its application to portfolio optimization

The aim of this paper is to propose a convex risk measure in the framework of fuzzy random theory and verify its advantage over the conventional variance approach. For this purpose, this paper defines the quadratic deviation (QD) of fuzzy random variable as the mathematical expectation of QDs of fuzzy variables. As a result, the new risk criterion essentially describes the variation of a fuzzy random variable around its expected value. For triangular and trapezoidal fuzzy random variables as well as their linear combinations, we establish the analytical expressions of their QDs, and obtain the desirable convexity about the analytical expressions with respect to critical parameters. To explore the practical value of the proposed QD, we apply it to a portfolio selection problem to quantify the investment risk, and develop three mean-QD models to find the optimal allocation of the fund in different risky securities. Due to the convexity of our QD, the original three mean-QD models can be turned into their equivalent convex parametric quadratic programming problems, which can be solved by conventional optimization methods. The computational results clearly demonstrate that our new QD significantly reduces the computational complexity that cannot be avoided when variance is used as a risk criterion. Finally, the numerical comparison between the proposed mean-QD model and mean-variance model is conducted to show the consistency between the optimal results in both techniques. Meanwhile, the comparison between the proposed QD, variance, spread, and second moment is made to summarize the similarities and differences between them, distinguish these four risk criteria and determine their respective application scopes in decision systems.

A Neural Network Architecture to Learn Explicit MPC Controllers from Data

We present a methodology to learn explicit Model Predictive Control (eMPC) laws from sample data points with tunable complexity. The learning process is cast in a special Neural Network setting where the coefficients of two linear layers and a parametric quadratic program (pQP) implicit layer are optimized to fit the training data. Thanks to this formulation, powerful tools from the machine learning community can be exploited to speed up the offline computations through high parallelization. The final controller can be deployed via low-complexity eMPC and the resulting closed-loop system can be certified for stability using existing tools available in the literature. A numerical example on the voltage-current regulation of a multicell DC-DC converter is provided, where the storage and on-line computational demands of the initial controller are drastically reduced with negligible performance impact.

Explicit spline-based continuous-time MPC: a study on design and performance

This paper presents a continuous-time model predictive control scheme based on B-spline functions used for signals and model approximation. The proposed controller offers two interesting advantages. First, it formulates the control signal as a continuous polynomial spline function, the nature of which is determined by its control polygon that is subject of optimization. Second, all continuous constraints assumed over prediction horizon are consistently transformed into constraints imposed on a finite number of elements of this control polygon. Using parametric quadratic programming we further show how to obtain an explicit representation of the proposed controller, which is known for its efficient online implementation. The featured simulation study demonstrates that by a suitable choice of number and position of knots of the spline function over the prediction horizon it is possible to substantially reduce the number of critical regions of the explicit controller while preserving control performance, and to mitigate the direct correlation between number of regions and chosen length of prediction horizon.

Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure

Portfolio selection is recognised as the birth-place of modern finance; portfolio optimisation has become a developed tool. However, efficient frontiers are piece-wisely made up by connected parabolic segments; such structure can be rendered only by parametric quadratic programming. Overlooking the structure can both get incomplete results and cause difficulties in e-constraint approaches or weighted-sums approaches. There has been no research to systematically parametrically compute efficient frontiers and report and analyse the structure up until now; in such an area, this article contributes to the literature. I utilise the software of parametric quadratic programming, set up practical portfolio selection models, build batches of 5-stock problems up to 1800-stock problems, analyse the structure, and report the findings. For example, the numbers of parabolic segments can quadratically increase with problem sizes, so fixed numbers of points are insufficient approximations of efficient frontiers. Contrary to common assumptions, an efficient frontier is not smooth in the presence of kinks. Moreover, I utilise the structure for rebalancing portfolios, propose two models to minimise rebalancing cost, transform them into linear programming or integer programming, and solve them. This article can help scholars and practitioners obtain a comprehensive picture of efficient frontiers and perceive the structure.

Complementarity-based nonlinear programming techniques for optimal mixing in gas networks

We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models already render small-size instances computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we show that classical stationarity concepts are meaningful for the resulting complementarity-based reformulation of the mixing equations. Further, we investigate in a numerical study the performance of this reformulation compared to a more compact complementarity-based one that does not feature such beneficial regularity properties. All computations are performed on publicly available data of real-world size problem instances from steady-state gas transport.

Boundary slippage modeling and optimization of hydrophobic tilting pad thrust bearing with elastic deformation

Abstract Boundary slippage modeling and optimization of the hydrophobic tilting pad thrust bearing with elastic deformation are conducted to obtain the optimal lubricating film geometry and power loss. The modified Reynolds equation combining with a nonlinear slip control equation of the limiting shear stress model is transformed to a series of linear complementary problems to be solved by the parametric quadratic programming technique. An optimization approach based on the Kriging surrogate model is developed to optimize the slip/no-slip configuration of the thrust pad. The effect of homogeneous slipping and heterogeneous slip/no-slipping coupled with elastic deformation on minimum film thickness and power loss is studied for the tilting pad thrust bearing with the flexible lining materials.

Parametrically computing efficient frontiers and reanalyzing efficiency-diversification discrepancies and naive diversification

Portfolio selection is recognized as the birth-place of modern finance. Weighted-sums methods or e-constraint methods are normally utilized for portfolio optimization, but the results are only approximations of efficient frontiers. One concern of portfolio selection is efficiency-diversification discrepancies that efficient frontiers lack diversification. Scholars typically analyze the discrepancies by using weighted-sums methods or e-constraint methods, studying only a specific portfolio, and utilizing small-scale portfolio selection. Some scholars find that portfolio selection is not consistently better than naive diversification. We utilize parametric quadratic programming, exhaustively sample US stocks, build batches of 5-stock problems up to 1800-stock problems, obtain the structure of (whole) efficient frontiers, and propose new diversification measures on the basis of the structure. We find that (1) setting upper bounds can be more effective in changing diversification status than setting right-hand sides or setting the numbers of constraints can, (2) portfolio selection can substantially outperform naive diversification at least in sample so the cost of naive diversification can be prohibitive, and (3) efficiency-diversification discrepancies can arise due to efficient frontiers’ nature of having relatively small numbers of stocks and cannot be easily reconciled.

PAL-Hom method for QP and an application to LP

In this paper, a proximal augmented Lagrangian homotopy (PAL-Hom) method for solving convex quadratic programming problems is proposed. This method takes the proximal augmented Lagrangian method as the outer iteration. To solve the proximal augmented Lagrangian subproblems, a homotopy method is presented as the inner iteration. The homotopy method tracks the piecewise-linear solution path of a parametric quadratic programming problem whose start problem takes an approximate solution as its solution and the target problem is the subproblem to be solved. To improve the performance of the homotopy method, the accelerated proximal gradient method is used to obtain a fairly good approximate solution that implies a good prediction of the optimal active set. Moreover, a sorting technique for the Cholesky factor update as well as an $$\varepsilon $$ -relaxation technique for checking primal-dual feasibility and correcting the active sets are presented to improve the efficiency and robustness of the homotopy method. Simultaneously, a proximal-point-based AL-Hom method which is shown to converge in finite number of steps, is applied to linear programming. Numerical experiments on randomly generated problems and the problems from the CUTEr and Netlib test collections, support vector machines (SVMs) and contact problems of elasticity demonstrate that PAL-Hom is faster than the active-set methods and the parametric active set methods and is competitive to the interior-point methods and the specialized algorithms designed for specific models (e.g., sequential minimal optimization method for SVMs).