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In this paper, we propose new linesearch-based methods for nonsmooth constrained optimization problems when first-order information on the problem functions is not available. In the first part, we describe a general framework for bound-constrained problems and analyze its convergence toward stationary points, using the Clarke--Jahn directional derivative. In the second part, we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth. In this case, we first split the constraints into two subsets: difficult general nonlinear constraints and simple bound constraints on the variables. Then, we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the bound-constrained minimization of the proposed exact penalty function. Finally, we use the framework developed for the bound-constrained case to solve the penalized problem. Moreover, we prove that every accumulation point, under standard assumptions on the search directions, of the generated sequence of iterates is a stationary point of the original constrained problem. In the last part of the paper, we report extended numerical results on both bound-constrained and nonlinearly constrained problems, showing that our approach is promising when compared to some state-of-the-art codes from the literature. [PUBLICATION ABSTRACT]

In this paper, an infeasible QP-free method without penalty function is proposed for inequality constrained optimization. We first compute a fundamental direction and then bend the search direction based on the constraint function and the Lagrange multiplier. Based on the modified nonmonotone filter technique, the acceptable criterion of trial points is relaxed and Maratos effects are avoided to a certain degree. At each iteration, only two or three systems of linear equations with the same coefficient are needed to solve to obtain the search direction. Under suitable conditions, the global convergence of the algorithm is proved without the strict complementarity conditions. In the end, some numerical results are reported.

In this paper, we propose a method to smooth the general lower-order exact penalty function for inequality constrained optimization. We prove that an approximation global solution of the original problem can be obtained by searching a global solution of the smoothed penalty problem. We develop an algorithm based on the smoothed penalty function. It is shown that the algorithm is convergent under some mild conditions. The efficiency of the algorithm is illustrated with some numerical examples.

Altitude test facilities for aero-engines employ multi-chamber, multi-valve intake systems that require effective decoupling and strong disturbance rejection during transient tests. This paper proposes a coordinated active disturbance rejection control (ADRC) scheme based on external penalty functions. The chamber pressure safety limit is formulated as an inequality-constrained optimization problem, and an exponential penalty together with a gradient based algorithm is designed for dynamic constraint relaxation, with guaranteed global convergence. A coordination term is then integrated into a distributed ADRC framework to yield a multi-valve coordinated ADRC controller, whose asymptotic stability is established via Lyapunov theory. Hardware-in-the-loop simulations using MATLAB/Simulink and a PLC demonstrate that, under ±3 kPa pressure constraints, the maximum engine inlet pressure error is 1.782 kPa (77.1% lower than PID control), and under an 80 kg/s2 flow-rate disturbance, valve oscillations decrease from ±27% to ±5%. These results confirm the superior disturbance rejection and decoupling performance of the proposed method.

Recent studies show that the filter method has good numerical performance for nonlinear complementary problems (NCPs). Their approach is to reformulate an NCP as a constrained optimization solved by filter algorithms. However, they can only prove that the iterative sequence converges to the KKT point of the constrained optimization. In this paper, we investigate the relation between the KKT point of the constrained optimization and the solution of the NCP. First, we give several sufficient conditions under which the KKT point of the constrained optimization is the solution of the NCP; second, we define regular conditions and regular point which include and generalize the previous results; third, we prove that the level sets of the objective function of the constrained optimization are bounded for a strongly monotone function or a uniform P-function; finally, we present some examples to verify the previous results.

For inequality constrained optimization problem, we first propose a new smoothing method to the lower order exact penalty function, and then show that an approximate global solution of the original problem can be obtained by solving a global solution of a smooth lower order exact penalty problem. We propose an algorithm based on the smoothed lower order exact penalty function. The global convergence of the algorithm is proved under some mild conditions. Some numerical experiments show the efficiency of the proposed method.

Penalty function is an important tool in solving many constrained optimization problems in areas such as industrial design and management. In this paper, we study exactness and algorithm of an objective penalty function for inequality constrained optimization. In terms of exactness, this objective penalty function is at least as good as traditional exact penalty functions. Especially, in the case of a global solution, the exactness of the proposed objective penalty function shows a significant advantage. The sufficient and necessary stability condition used to determine whether the objective penalty function is exact for a global solution is proved. Based on the objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the objective penalty function is proved for a local solution. An algorithm is presented in the paper in finding a local solution, with its convergence proved under some conditions. Finally, numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm.

This paper develops a reduced Hessian method for solving inequality constrained optimization problems. At each iteration, the proposed method solves a quadratic subproblem which is always feasible by introducing a slack variable to generate a search direction and then computes the steplength by adopting a standard line search along the direction through employing the l ∞ penalty function. And a new update criterion is proposed to generate the quasi-Newton matrices, whose dimensions may be variable, approximating the reduced Hessian of the Lagrangian. The global convergence is established under mild conditions. Moreover, local R-linear and superlinear convergence are shown under certain conditions.

Outer approximations are a well known technique for solving semiinfinite optimization problems. We show that a straightforward adaptation of this technique results in a new, external , active-set strategy that can easily be added to existing software packages for solving nonlinear programming problems with a large number of inequality constraints. Our external active-set strategy is very easy to implement, and, as our numerical results show, it is particularly effective when applied to discretized semiinfinite optimization or state-constrained optimal control problems. Its effects can be spectacular, with reductions in computing time that become progressively more pronounced as the number of inequalities is increased.

In this study, the frequency spectrum of a pulse for ultra wideband impulse radio systems is assumed to be the frequency response of a real valued causal rational infinite impulse response (IIR) filter. This pulse design problem is formulated as a functional inequality constrained optimisation problem. The objective of this optimisation problem is to minimise the sum of the total passband and stopband ripple energy of the pulse. Also, the optimisation problem is subject to the stability condition of the IIR filter as well as to the specifications defined on both the maximum passband and stopband ripple magnitudes and that on the maximum time domain ripple magnitude of the pulse. If the feasible set of the optimisation problem is non-empty, then the designed pulse is guaranteed to be satisfied the specifications suggested by the Federal Communications Commission. Computer numerical simulation results show that the energy compaction performance defined in the frequency domain of the authors’ designed pulse outperforms that of existing pulses. Since the frequency spectrum of the pulse is the same as the frequency response of the designed IIR filter, the pulse can be generated via an excitation of a simple circuit.