Inequality Constrained Optimization Problems 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.

An exact penalty function approach for inequality constrained optimization problems based on a new smoothing technique

Exact penalty methods are one of the effective tools to solve nonlinear programming problems with inequality constraints. In this study, a new class of exact penalty functions is defined and a new family of smoothing techniques to exact penalty functions is introduced. Error estimations are presented among the original, non-smooth exact penalty and smoothed exact penalty problems. It is proved that an optimal solution of smoothed penalty problem is an optimal solution of original problem. A smoothing penalty algorithm based on the the new smoothing technique is proposed and the convergence of the algorithm is discussed. Finally, the efficiency of the algorithm on some numerical examples is illustrated.

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.

A combined experimental and analytical method to determine the EHL friction force distribution between rollers and outer raceway in a cylindrical roller bearing

Friction force is a crucial factor causing power loss and fatigue spalling of rolling element bearings. A combined experimental and analytical method is proposed to quantitatively determine the elastohydrodynamic lubrication (EHL) friction force distribution between rollers and outer raceway in a cylindrical roller bearing (CRB). An experimental system with the instrumented bearing and housing was developed for measuring radial load distribution and friction torque of bearings. A simplified model of friction force expressed by dimensionless speed, load, and material parameters was given. An inequality constrained optimization problem was established and solved by using an experimental data-driven learning algorithm for determining the uncertain parameters in the model. The effect of speed, load, and lubricant property on friction force and friction coefficient was discussed.

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.

A General Solution for the Errors in Variables (EIV) Model with Equality and Inequality Constraints

Targeting the adjustment of the errors-in-variables (EIV) model with equality and inequality constraints, a general solution that is similar to the classical least square adjustment is proposed based on the penalty function and the weight in measurement. Firstly, we take the equality constraints as inequality constraints that do not satisfy the constraint conditions and construct the penalty functions of equality and inequality constraints, respectively. Thus, the inequality constrained optimization problem is transformed into an unconstrained optimization problem. Then the detailed calculation formula and approximate accuracy evaluation formula of the general solution are deduced. The iteration formula of the general solution is easy regarding comprehension and applicable in implementation. It can not only solve the EIV model with equality and inequality constraints respectively, but also address the EIV model with equality and inequality constraints simultaneously. In addition, it can promote the Gauss–Markov (G-M) model with equality and inequality constraints. Finally, three examples (i.e., equality constraints, inequality constraints and those with equality and inequality constraints) are validated, indicating that the general solution is effective and feasible. The results show that the general solution is effective and feasible.

Smoothing Approximation to the New Exact Penalty Function with Two Parameters

In this paper, we propose a new non-smooth penalty function with two parameters for nonlinear inequality constrained optimization problems. And we propose a twice continuously differentiable function which is smoothing approximation to the non-smooth penalty function and define the corresponding smoothed penalty problem. A global solution of the smoothed penalty problem is proved to be an approximation global solution of the non-smooth penalty problem. Based on the smoothed penalty function, we develop an algorithm and prove that the sequence generated by the algorithm can converge to the optimal solution of the original problem.

Infinite Impulse Response Filter Bank Based Graphic Equalizer Design via Functional Inequality Constrained Optimization and Genetic Algorithm

It is worth noting that a graphical equalizer is a linear time-invariant system. Hence, its input-output relationship can be modeled by a transfer function. The graphical equalizer is realized by a parallel connection of a set of filters called a filter bank. Each filter in the filter bank is designed to equalize a certain frequency band of the audio signal. There are two types of graphic equalizers. The first type of graphical equalizers is based on a finite impulse response (FIR) filter bank. However, it requires very high order FIR filters to achieve a certain degree of equalization precision. As a result, the required computational power is very high. Also, the output of the graphical equalizer is suffered from a very long time delay. The second type of graphical equalizers is based on an infinite impulse response (IIR) filter bank. It can address the above issues. That is, its implementation cost is low. To design each IIR filter, the total absolute error between the ideal equalization and the actual equalization in the passband of the IIR filter is minimized subject to the constraints imposed on the stability criterion of the IIR filter as well as on the maximum modulus error between the ideal equalization and the actual equalization in the corresponding passband. This design problem is a functional inequality constrained optimization problem which consists of an infinite number of inequality constraints. By performing the downsampling in the frequency domain, the infinite constrained optimization problem becomes the finite constrained optimization problem. By using the genetic algorithm to find the solution of this optimization problem, the coefficients of all the IIR filters are obtained. The computer numerical simulation results show that the equalizer designed by our proposed method yields a good equalization performance.

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.

The Pareto Tracer for General Inequality Constrained Multi-Objective Optimization Problems

Problems where several incommensurable objectives have to be optimized concurrently arise in many engineering and financial applications. Continuation methods for the treatment of such multi-objective optimization methods (MOPs) are very efficient if all objectives are continuous since in that case one can expect that the solution set forms at least locally a manifold. Recently, the Pareto Tracer (PT) has been proposed, which is such a multi-objective continuation method. While the method works reliably for MOPs with box and equality constraints, no strategy has been proposed yet to adequately treat general inequalities, which we address in this work. We formulate the extension of the PT and present numerical results on some selected benchmark problems. The results indicate that the new method can indeed handle general MOPs, which greatly enhances its applicability.

Research on Hyperbola Fitting Algorithm for Turbulence Level Measurement Test Data

Hyperbola fitting of test data is an extremely important process in turbulence level measurement test in wind tunnels. The solution of the overdetermined equations (SOE) method is often used to solve hyperbola fitting parameters to obtain turbulence level. However, due to unsteady flow characteristics, the SOE method often results in overfitting phenomena, which makes it impossible to solve turbulence level accurately. This paper proposes using the constrained least-squares (CLS) method to convert the problem of hyperbola fitting of test data into the inequality constrained optimization problem and then using the Lagrange programming neural network (LPNN) method to solve turbulence level iteratively. The stability of the LPNN method is analysed, and three sets of typical turbulence level measurement test data are processed using the LPNN method. The results verify the feasibility of applying the LPNN method to iteratively solve the turbulence level of wind tunnels.

A sequential quadratic programming algorithm without a penalty function, a filter or a constraint qualification for inequality constrained optimization

This paper discusses a kind of nonlinear inequality constrained optimization problems without any constraint qualification. A new sequential quadratic programming algorithm for such problems is proposed, whose important features are as follows: (i) a new relaxation technique for the linearized constraints of the quadratic programming subproblem is introduced, which guarantees that the subproblem is always consistent and generates a favourable search direction; (ii) a weaker positive-definiteness assumption on the quadratic coefficient matrices is presented; (iii) a slightly new line search is adopted, where neither a penalty function nor a filter is used; (iv) an associated acceptable termination rule is introduced; (v) the finite convergence of the algorithm is proved. Furthermore, the numerical results on a collection of CUTE test problems show that the proposed algorithm is promising.

Coupling Elephant Herding with Ordinal Optimization for Solving the Stochastic Inequality Constrained Optimization Problems

The stochastic inequality constrained optimization problems (SICOPs) consider the problems of optimizing an objective function involving stochastic inequality constraints. The SICOPs belong to a category of NP-hard problems in terms of computational complexity. The ordinal optimization (OO) method offers an efficient framework for solving NP-hard problems. Even though the OO method is helpful to solve NP-hard problems, the stochastic inequality constraints will drastically reduce the efficiency and competitiveness. In this paper, a heuristic method coupling elephant herding optimization (EHO) with ordinal optimization (OO), abbreviated as EHOO, is presented to solve the SICOPs with large solution space. The EHOO approach has three parts, which are metamodel construction, diversification and intensification. First, the regularized minimal-energy tensor-product splines is adopted as a metamodel to approximately evaluate fitness of a solution. Next, an improved elephant herding optimization is developed to find N significant solutions from the entire solution space. Finally, an accelerated optimal computing budget allocation is utilized to select a superb solution from the N significant solutions. The EHOO approach is tested on a one-period multi-skill call center for minimizing the staffing cost, which is formulated as a SICOP. Simulation results obtained by the EHOO are compared with three optimization methods. Experimental results demonstrate that the EHOO approach obtains a superb solution of higher quality as well as a higher computational efficiency than three optimization methods.

Null space gradient flows for constrained optimization with applications to shape optimization

The purpose of this article is to introduce a gradient-flow algorithm for solving equality and inequality constrained optimization problems, which is particularly suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita (Math. Program. 18 (1980) 155–168) for equality constrained problems: the search direction is a combination of a null space step and a range space step, aiming to decrease the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. In the literature, a common practice consists in reducing inequality constraints to equality constraints by the introduction of additional slack variables. Here, we rather solve their local combinatorial character by computing the projection of the gradient of the objective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem whose size equals the number of active or violated constraints. The solution to this problem allows to identify the inequality constraints to which the optimization trajectory should remain tangent. Our second contribution is a formulation of our gradient flow in the context of – infinite-dimensional – Hilbert spaces, and of even more general optimization sets such as sets of shapes, as it occurs in shape optimization within the framework of Hadamard’s boundary variation method. The cornerstone of this formulation is the classical operation of extension and regularization of shape derivatives. The numerical efficiency and ease of implementation of our algorithm are demonstrated on realistic shape optimization problems.

A modified search direction method for inequality constrained optimization problems using the singular-value decomposition of normalized response gradients

Motivated by the applications of the node-based shape optimization problems, where various response evaluations are often considered in constrained optimization, we propose a modified search direction method for optimization design updates. There exist numerous methods for solving constrained optimization problems. The methods of the class active-set strategy try to travel along the active constraints to solve constrained optimization problems. Contrary to the active-set methods, the algorithms of the class interior-point method reach an optimal solution by traversing the interior of the feasible domain. This characteristic is considered to be beneficial for shape optimization because the usual zig-zagging behavior when traveling along the active constraints is avoided. However, the interior-point methods require the solution of a Newton problem in each iteration step, which is generally considered to be difficult for the shape optimization problem. In the present work, we propose a modified search direction method that applies the singular-value decomposition method for the normalized objective and constraint gradients. The modified search direction is a descent direction of the objective function. Using this search direction for the design update in each iteration, we observe that the centrality conditions for the interior-point method are approached iteratively. We are able to achieve a local minimum by traversing the interior of the feasible domain by only using the gradient information of the objective and constraint functions. The results are shown first with analytical 2D problems, and then the results of shape optimization problems with a large number of design variables are discussed. To robustly deal with complex geometries, the Vertex Morphing method is used.

2-factor Newton method for solving the constrained optimization problem with the singular Kuhn—Tucker system

A new method for solving the inequality constrained optimization problem is proposed for the case when the system of necessary optimality conditions of Kuhn—Tucker is degenerate. This situation occurs for example in the case when strict complementarity conditions fails in solution point. The reduction of the inequalities con- strained optimization problem to the equalities constrained problem is substantiated and the use of a new 2-fac- tor Newton method for the effective solution of the obtained degenerate system of optimality conditions is shown.

On second-order Fritz John type optimality conditions for a class of differentiable optimization problems

In this paper, we focus on the inequality constrained optimization problems, where the involved functions belong to the class , i.e. functions which have Lipschitz continuous gradient mappings. We propose a second-order mean value inequality for a function in terms of its limiting second-order subdifferential. By virtue of this inequality, using the second-order tangent set and the asymptotic second-order tangent cone, we establish the second-order Fritz John type necessary and sufficient optimality conditions for differentiable optimization problems with inequality and set constraints.

Intrinsic Formulation of KKT Conditions and Constraint Qualifications on Smooth Manifolds

Karush-Kuhn-Tucker (KKT) conditions for equality and inequality constrained optimization problems on smooth manifolds are formulated. Under the Guignard constraint qualification, local minimizers are shown to admit Lagrange multipliers. The linear independence, Mangasarian-Fromovitz, and Abadie constraint qualifications are also formulated, and the chain "LICQ implies MFCQ implies ACQ implies GCQ" is proved. Moreover, classical connections between these constraint qualifications and the set of Lagrange multipliers are established, which parallel the results in Euclidean space. The constrained Riemannian center of mass on the sphere serves as an illustrating numerical example.

A new filter QP-free method for the nonlinear inequality constrained optimization problem

In this paper, a filter QP-free infeasible method with nonmonotone line search is proposed for minimizing a smooth optimization problem with smooth inequality constraints. This proposed method is based on the solution of nonsmooth equations, which are obtained by the Lagrangian multiplier method and the function of the nonlinear complementarity problem for the Karush–Kuhn–Tucker optimality conditions. Especially, each iteration of this method can be viewed as a perturbation of a Newton or quasi-Newton iteration on both the primal and dual variables for the solution of the Karush–Kuhn–Tucker optimality conditions. What is more, it is considered to use the function of the nonlinear complementarity problem in the filter, which makes the proposed algorithm avoid the incompatibility. Then the global convergence of the proposed method is given. And under some mild conditions, the superlinear convergence rate can be obtained. Finally, some preliminary numerical results are shown to illustrate that the proposed filter QP-free infeasible method is quite promising.