Inequality-constrained Optimization Research Articles

Optimal control using flux potentials: A way to construct bound-preserving finite element schemes for conservation laws

To ensure preservation of local or global bounds for numerical solutions of conservation laws, we constrain a baseline finite element discretization using optimization-based (OB) flux correction. The main novelty of the proposed methodology lies in the use of flux potentials as control variables and targets of inequality-constrained optimization problems for numerical fluxes. In contrast to optimal control via general source terms, the discrete conservation property of flux-corrected finite element approximations is guaranteed without the need to impose additional equality constraints. Since the number of flux potentials is less than the number of fluxes in the multidimensional case, the potential-based version of optimal flux control involves fewer unknowns than direct calculation of optimal fluxes. We show that the feasible set of a potential-state potential-target (PP) optimization problem is nonempty and choose a primal–dual Newton method for calculating the optimal flux potentials. The results of numerical studies for linear and nonlinear conservation laws in 2D demonstrate the superiority of the new OB-PP algorithms to closed-form flux limiting under worst-case assumptions.

Model Predictive Control of Autonomous Vehicles With Integrated Barriers Using Occupancy Grid Maps

Nonlinear model predictive control (NMPC) is an efficient and proven method for optimization-based autonomous vehicle motion planning. Among the various approaches, the iterative linear quadratic regulator, a differential dynamic programming variant, is a well-known efficient nonlinear optimization method. In safety-critical control systems, controllers should address inequality-constrained optimization problems. In this letter, we design a single unified constraint using an occupancy grid map. We convert this inequality-constrained optimization problem into an unconstrained optimization problem by appending a single integrated discrete barrier state to the system model. This approach simplifies complex motion planning problems and reduces computational costs. In the proposed method, we first discretize the surrounding environment with an occupancy grid map and design a single constraint that ensures that only cells with values less than a predefined threshold can be traversed by the ego vehicle. Then, we define a single integrated discrete barrier state to introduce this constraint into the motion planning algorithm. The proposed method, a penalty method, and the augmented Lagrangian method are tested on a real-time software-in-the-loop simulation using CarMaker and ROS. The simulation results of pop-up obstacle avoidance scenarios show the benefits of the proposed method, such as reduced time costs and increased robustness.

Constrained shuffled complex evolution algorithm and its application in the automatic calibration of Xinanjiang model

The Shuffled Complex Evolution—University of Arizona (SCE-UA) is a classical algorithm in the field of hydrology and water resources, but it cannot solve constrained optimization problems directly. Using penalty functions has been the preferred method to handle constraints, but the appropriate selection of penalty parameters and penalty functions can be challenging. To enhance the universality of the SCE-UA, we propose the Constrained Shuffled Complex Evolution Algorithm (CSCE) to conveniently and effectively solve inequality-constrained optimization problems. Its performance is compared with the SCE-UA using the adaptive penalty function (SCEA) on 14 test problems with inequality constraints. It is further compared with seven other algorithms on two test problems with low success rates. To demonstrate its effect in hydrologic model calibration, the CSCE is applied to the parameter optimization of the Xinanjiang (XAJ) model under synthetic data and observed data. The results indicate that the CSCE is more advantageous than the SCEA in terms of the success rate, stability, feasible rate, and convergence speed. It can guarantee the feasibility of the solution and avoid the problem of deep soil tension water capacity (WDM)<0 in the optimization process of the XAJ model. In the case of synthetic data, the CSCE can accurately find the theoretical optimal parameters of the XAJ model under the given constraints. In the case of observed data, the XAJ model optimized by the CSCE can effectively simulate the hourly rainfall-runoff events of the Hexi Basin and achieves mean Nash efficiency coefficients greater than 0.75 in the calibration period and the validation period.

On some variational inequality-constrained control problems

In this paper, by considering some properties associated with scalar functionals of multiple-integral type, we study the well-posedness and generalized well-posedness for a new variational inequality-constrained optimization problems By using the set of approximating solutions, we state some characterization theorems on well-posedness and generalized well-posedness. Also, in order to validate the derived results, some examples are given.

The energy release rate for non-penetrating crack in poroelastic body by fluid-driven fracture

A new class of constrained variational problems, which describe fluid-driven cracks (that are pressurized fractures created by pumping fracturing fluids), is considered within the nonlinear theory of coupled poroelastic models stated in the incremental form. The two-phase medium is constituted by solid particles and fluid-saturated pores; it contains a crack subjected to non-penetration condition between the opposite crack faces. The inequality-constrained optimization is expressed as a saddle-point problem with respect to the unknown solid phase displacement, pore pressure, and contact force. Applying the Lagrange multiplier approach and the Delfour–Zolésio theorem, the shape derivative for the corresponding Lagrangian function is derived using rigorous asymptotic methods. The resulting formula describes the energy release rate under irreversible crack perturbations, which is useful for application of the Griffith criterion of quasi-static fracture.

Uncertainty-driven active view planning in feature-based monocular vSLAM

Traditional feature-based monocular visual simultaneous localization and mapping (vSLAM) methods suffer from frequent tracking failure in low-texture scenes. Although tracking stability can be guaranteed by actively adjusting the camera’s orientation to track known landmarks, it can easily lead to the problem of over-exploitation. This means that instead of discovering new landmarks, the camera focuses on an area that has already been observed, which is not conducive to fully exploring unknown environments. To address this problem, an uncertainty-driven active view planning framework is proposed to actively adjust the orientation of the monocular camera equipped on a three degree of freedom (3-DoF) pan–tilt. As a result, a trade-off between exploitation, i.e., making full use of known information, and exploration, i.e., obtaining more information of unknown environments can be achieved. First, a novel landmark uncertainty model is established to represent the uncertainty of environmental information. Second, the trade-off problem is formulated as an inequality-constrained optimization mathematical model, whose objective function is related to landmark uncertainty. Last, Karush–Kuhn–Tucker (KKT) conditions are utilized to solve the optimization problem. Experimental results on a publicly available monocular dataset and in a real-world environment show that this framework reduces the rate of tracking failure by 50% on average. The localization error is also reduced by 0.07 m for translation and 0.004 rad/m for rotation on average. Meanwhile, the number of reconstructed landmarks increases by 17.86% on average, which indicates an appropriate trade-off between exploitation and exploration.

An Interior-Point Approach for Solving Risk-Averse PDE-Constrained Optimization Problems with Coherent Risk Measures

The prevalence of uncertainty in models of engineering and the natural sciences necessitates the inclusion of random parameters in the underlying partial differential equations (PDEs). The resulting decision problems governed by the solution of such random PDEs are infinite dimensional stochastic optimization problems. In order to obtain risk-averse optimal decisions in the face of such uncertainty, it is common to employ risk measures in the objective function. This leads to risk-averse PDE-constrained optimization problems. We propose a method for solving such problems in which the risk measures are convex combinations of the mean and conditional value-at-risk (CVaR). Since these risk measures can be evaluated by solving a related inequality-constrained optimization problem, we suggest a log-barrier technique to approximate the risk measure. This leads to a new continuously differentiable convex risk measure: the log-barrier risk measure. We show that the log-barrier risk measure fits into the setting of optimized certainty equivalents of Ben-Tal and Teboulle and the expectation quadrangle of Rockafellar and Uryasev. Using the differentiability of the log-barrier risk measure, we derive first-order optimality conditions reminiscent of classical primal and primal-dual interior-point approaches in nonlinear programming. We derive the associated Newton system, propose a reduced symmetric system to calculate the steps, and provide a sufficient condition for local superlinear convergence in the continuous setting. Furthermore, we provide a $\Gamma$-convergence result for the log-barrier risk measures to prove convergence of the minimizers to the original nonsmooth problem. The results are illustrated by a numerical study.

A volume-based approach to phase equilibrium calculations at pressure and temperature specifications

Phase equilibrium calculations at pressure and temperature specifications, consisting in the minimization of the Gibbs free energy with respect to mole numbers, are the most commonly used and are well documented in the literature. A very attractive alternative is given by the volume-based calculations, in which the volume and mole numbers, which are the natural variables for pressure explicit equations of state (EoS) are treated as independent variables; in this case pressure equality is an additional equation for each phase and there is no need to solve the EoS for volume. The problem is a bound- and linear inequality-constrained optimization problem; the objective function is a formal Gibbs free energy. A major disadvantage is that, unlike in conventional calculations, the successive substitution method cannot be used, thus robust modified Newton iterations are required. Using the block structure of the Hessian matrix, a link is established between conventional and volume-based Newton iterations and it is shown that the latter are inherently slower than the former due to a lower implicitness level in the Hessian. A modified Cholesky factorization (to ensure a descent direction) and a two-stage line search procedure (ensuring that iterates remain in the feasible domain and the objective function is decreasing between two iterations) are used in Newton iterations. Numerical experiments carried out on several test mixtures show that two-phase volume-based flashes initialized from the ideal equilibrium constants pose systematic convergence problems near phase boundaries, while the algorithm is rapid and robust when initial guesses from stability testing are used. The proposed algorithm is not dependent on the thermodynamic model and any pressure explicit EoS can be used, provided the required partial derivatives are available.

Synchronized flux limiting for gas dynamics variables

This work addresses the design of failsafe flux limiters for systems of conserved quantities and derived variables in numerical schemes for the equations of gas dynamics. Building on Zalesak's multidimensional flux-corrected transport (FCT) algorithm, we construct a new positivity-preserving limiter for the density, total energy, and pressure. The bounds for the underlying inequality constraints are designed to enforce local maximum principles in regions of strong density variations and become less restrictive in smooth regions. The proposed approach leads to closed-form expressions for the synchronized correction factors without the need to solve inequality-constrained optimization problems. A numerical study is performed for the compressible Euler equations discretized using a finite element based FCT scheme.

Analytical Approach in Computing Nash Equilibrium for Oligopolistic Competition of Transmission-Constrained GENCOs

This paper proposes an analytical solution for bidding strategy problem of transmission-constrained generating companies (GENCOs) in an hour-ahead electricity market. Here, for modeling a bidding strategy problem under a supply function equilibrium model, a bilevel programming (BLP) method is used. In the upper level, the profit of an intended GENCO is maximized, and in the lower level, the independent system operator clears the market via solving a bid-based security-constrained economic dispatch (SCED), in which SCED can be parameterized by GENCOs' bidding strategies. We apply an active-set method to solve a parametric SCED problem. The idea of implementing the active-set method is based on the fact that the optimal solution of a parametric quadratic programming (QP) problem with linear equality constraints can be determined directly. We also decompose generation scheduling, locational marginal prices, and consequently GENCOs' profits, in terms of GENCOs' bidding strategies. In this situation, the BLP transforms to a single-level inequality-constrained optimization, which maximizes GENCO's profit subject to inactive constraints of the lower level. In addition, GENCOs' strategic interactions in oligopolistic marketplace are investigated by using a game theoretic framework, and optimal bidding strategies are determined then by finding the Nash equilibrium of GENCOs' noncooperative competition. Two case studies are provided to illustrate the proposed method; in addition, a numerical method has been conducted to validate the results. The results clarify the impact of each GENCO's bidding strategy on the power flows of transmission lines.

3-1 Piecewise NCP Function for New Nonmonotone QP-Free Infeasible Method

<div class=""abs_img""><img src=""[disp_template_path]/JRM/abst-image/00260005/04.jpg"" width=""300"" />Algorithm flow chart</div> We propose a nonmonotone QP-free infeasible method for inequality-constrained nonlinear optimization problems based on a 3-1 piecewise linear NCP function. This nonmonotone QP-free infeasible method is iterative and is based on nonsmooth reformulation of KKT first-order optimality conditions. It does not use a penalty function or a filter in nonmonotone line searches. This algorithm solves only two systems of linear equations with the same nonsingular coefficient matrix, and is implementable and globally convergent without a linear independence constraint qualification or a strict complementarity condition. Preliminary numerical results are presented. </span>

A QP-free algorithm of quasi-strongly sub-feasible directions for inequality constrained optimization

In this paper, combining the method of quasi-strongly sub-feasible directions (MQSSFD) and the working set technique, a new QP-free algorithm with an arbitrary initial iteration point for solving inequality constrained optimization is proposed. At each iteration, the algorithm solves only two systems of linear equations with a same uniformly nonsingular coefficient matrix to obtain the search direction. Furthermore, the positive definiteness assumption on the Hessian estimate is relaxed. Under some necessary assumptions, the new algorithm not only possesses global and strong convergence, but also ensures that the iteration points can get into the feasible set after finite iterations. Finally, a series of preliminary numerical results are reported to show that the algorithm is promising.

An infeasible QP-free algorithm without a penalty function or a filter for nonlinear inequality-constrained optimization

In this paper, we present an infeasible quadratic programming-free method without using a penalty function or a filter for inequality-constrained optimization. At each iteration, three linear equations with the same coefficient matrix are solved. Furthermore, the nearly active set technique is used to reduce the amount of computational work. Under mild conditions, we prove that the proposed method has global and superlinear local convergence. At last, preliminary numerical results and comparison are reported.

A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization

A new parallel variable distribution algorithm based on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved.

Bounds on Eigenvalues of Matrices Arising from Interior-Point Methods

Interior-point methods feature prominently among numerical methods for inequality-constrained optimization problems, and involve the need to solve a sequence of linear systems that typically become increasingly ill-conditioned with the iterations. To solve these systems, whose original form has a nonsymmetric $3\times 3$ block structure, it is common practice to perform block elimination and either solve the resulting reduced saddle-point system, or further reduce the system to the Schur complement and apply a symmetric positive definite solver. In this paper we use energy estimates to obtain bounds on the eigenvalues of the matrices, and conclude that the original unreduced matrix has more favorable eigenvalue bounds than the alternative reduced versions. Our analysis includes regularized variants of those matrices that do not require typical regularity assumptions.

A Journey Through the 'Mathematical Underworld' of Portfolio Optimization

* It has been estimated that the current size of the asset management industry is approximately US$58 trillion.* Portfolio optimization is one of the problems most frequently encountered by financial practitioners. It appears in various forms in the context of Trading, Risk Management and Capital Allocation.* The Critical Line Algorithm (CLA) is the only algorithm specifically designed for inequality-constrained portfolio optimization problems, which guarantees that the exact solution is found after a predefined number of iterations.* Surprisingly, open-source implementations of CLA in a scientific language appear to be inexistent or unavailable.* The lack of publicly available CLA software, commercially or open-source, means that trillions of dollars are likely to be suboptimally allocated as a result of practitioners using general-purpose quadratic optimizers.* We believe that a large amount of financial firms and practitioners will benefit from our robust implementation of CLA in a scientific language.The code is available in the author's website.

An Open-Source Implementation of the Critical-Line Algorithm for Portfolio Optimization

Portfolio optimization is one of the problems most frequently encountered by financial practitioners. To our knowledge, the Critical Line Algorithm (CLA) is the only algorithm specifically designed for inequality-constrained portfolio optimization problems, which guarantees that the exact solution is found after a given number of iterations. Furthermore, CLA does not only compute a single portfolio, but it derives the entire efficient frontier. In the context of portfolio optimization problems, this approach is clearly more adequate than generic-purpose quadratic programming algorithms. The main goal of this paper is to fill a gap in the literature by providing a well-documented, step-by-step open-source implementation of CLA in a scientific language. The code is implemented as a Python class object, which allows it to be imported like any other Python module, and integrated seamlessly with pre-existing code. We discuss the logic behind CLA following the algorithm’s decision flow. In addition, we have developed several utilities that facilitate the answering of recurrent practical problems. We hope that this publication will offer a better alternative to financial practitioners, many of whom are currently relying on generic-purpose optimizers which often deliver suboptimal solutions. The source code can be downloaded at the authors' websites.

A Generalized Gradient Projection Filter Algorithm for Inequality Constrained Optimization

A generalized gradient projection filter algorithm for inequality constrained optimization is presented. It has three merits. The first is that the amount of computation is lower, since the gradient matrix only needs to be computed one time at each iterate. The second is that the paper uses the filter technique instead of any penalty function for constrained programming. The third is that the algorithm is of global convergence and locally superlinear convergence under some mild conditions.

Implementation of numerical integration schemes for the simulation of magnetic SMA constitutive response

Several constitutive models for magnetic shape memory alloys (MSMAs) have been proposed in the literature. The implementation of numerical integration schemes, which allow the prediction of constitutive response for general loading cases and ultimately the incorporation of MSMA response into numerical solution algorithms for fully coupled magneto-mechanical boundary value problems, however, has received only very limited attention. In this work, we establish two algorithmic implementations of the internal variable model for MSMAs proposed in (Kiefer and Lagoudas 2005 Phil. Mag. Spec. Issue: RecentAdv. Theor. Mech. 85 4289–329, Kiefer and Lagoudas 2009 J. Intell. Mater.Syst. 20 143–70), where we restrict our attention to pure martensitic variant reorientation to limit complexity. The first updating scheme is based on the numerical integration of the reorientation strain evolution equation and represents a classical predictor–corrector-type general return mapping algorithm. In the second approach, the inequality-constrained optimization problem associated with internal variable evolution is converted into an unconstrained problem via Fischer–Burmeister complementarity functions and then iteratively solved in standard Newton–Raphson format. Simulations are verified by comparison to closed-form solutions for experimentally relevant loading cases.

Formulation, analysis and numerical study of an optimization-based conservative interpolation (remap) of scalar fields for arbitrary Lagrangian–Eulerian methods

We develop and study the high-order conservative and monotone optimization-based remap (OBR) of a scalar conserved quantity (mass) between two close meshes with the same connectivity. The key idea is to phrase remap as a global inequality-constrained optimization problem for mass fluxes between neighboring cells. The objective is to minimize the discrepancy between these fluxes and the given high-order target mass fluxes, subject to constraints that enforce physically motivated bounds on the associated primitive variable (density). In so doing, we separate accuracy considerations, handled by the objective functional, from the enforcement of physical bounds, handled by the constraints. The resulting OBR formulation is applicable to general, unstructured, heterogeneous grids. Under some weak requirements on grid proximity, but not on the cell types, we prove that the OBR algorithm is linearity preserving in one, two and three dimensions. The paper also examines connections between the OBR and the recently proposed flux-corrected remap (FCR), Liska et al. [1]. We show that the FCR solution coincides with the solution of a modified version of OBR (M-OBR), which has the same objective but a simpler set of box constraints derived by using a “worst-case” scenario. Because M-OBR (FCR) has a smaller feasible set, preservation of linearity may be lost and accuracy may suffer for some grid configurations. Our numerical studies confirm this, and show that OBR delivers significant increases in robustness and accuracy. Preliminary efficiency studies of OBR reveal that it is only a factor of 2.1 slower than FCR, but admits 1.5 times larger time steps.