Search

This paper presents a novel nonsmooth objective penalty function for inequality constrained optimization problems. A modified flattened aggregate function, which is a smooth approximation of the max-value function, is discussed. Then, the smooth objective penalty function that contains the flattened aggregate function is proposed, and the exactness of the new function is studied. Based on this, an objective penalty function algorithm is proposed and its convergence is proven under mild conditions. Because of the flattened aggregate function, the gradient computation can usually be greatly reduced for problems with many constraints. Numerical experiments are included to illustrate the efficiency of the new algorithm through a series of numerical examples, especially for solving problems with many constraints.

A new exact projective penalty method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing its value at the projection of an infeasible point on the feasible set with the distance to the projection. Beside Euclidean projections, also a pointed projection in the direction of some fixed internal feasible point can be used. The equivalence means that local and global minimums of the problems coincide. Nonconvex sets with multivalued Euclidean projections are admitted, and the objective function may be lower semicontinuous. The particular case of convex problems is included. The obtained unconstrained or box constrained problem is solved by a version of the branch and bound method combined with local optimization. In principle, any local optimizer can be used within the branch and bound scheme but in numerical experiments sequential quadratic programming method was successfully used. So the proposed exact penalty method does not assume the existence of the objective function outside the allowable area and does not require the selection of the penalty coefficient. [ABSTRACT FROM AUTHOR]

In this two-part study, we develop a general theory of the so-called exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. In contrast to traditional nonsmooth exact penalty functions, these augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, which makes them especially appealing for applications in numerical optimization. Our aim is to present a detailed study of various theoretical properties of exact augmented Lagrangians and discuss several applications of these functions to constrained variational problems, problems with PDE constraints, and optimal control problems. The first paper is devoted to a theoretical analysis of an exact augmented Lagrangian for optimization problems in Hilbert spaces. We obtain several useful estimates of this augmented Lagrangian and its gradient, and present several types of sufficient conditions for KKT-points of a constrained problem corresponding to locally/globally optimal solutions to be local/global minimizers of the exact augmented Lagrangian. [ABSTRACT FROM AUTHOR]

• A differential exact penalty function exploiting the topology optimization penalty principle is proposed. • New active and loss functions are devised to treat the violation of equality and inequality constraints. • The derived optimization conditions without dual and slack variables coincide with the canonical KKT conditions. • The derivatives of the proposed loss functions can be interpreted as the traditional Lagrange multipliers. • The appearance of the first order partial derivatives in the Hessian matrix is unveiled. A smooth penalty function method which does not involve dual and slack implicit variables to formulate constrained optimization conditions is proposed. New active and loss functions are devised to independently and adaptively handle the violation of each constraint function. A single unconstrained minimization of the proposed penalty function produces a solution to the original optimization problem. The constrained optimization conditions depending only on the primal variable are derived and reduce to the canonical Karush-Kuhn-Tucker(KKT) conditions in a much more general framework. The derivative of the proposed loss functions with respect to constraint function can be interpreted as the Lagrange multiplier of the traditional Lagrangian function. The appearance of the first-order Hessian information is unveiled. This is in contrast to the existing works where the classical Lagrange Hessian contains only second order derivatives. Finally, some numerical examples including medium-scale stress constrained topology optimization and scenario based reliability design problems are presented to demonstrate the effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

In this paper, a class of smooth penalty functions is proposed for constrained optimization problem. It is put forward based on L p , a smooth function of a class of exact penalty function ℓ p p ∈ (0 , 1 ] . Based on the class of penalty functions, a penalty algorithm is presented. Under the very weak condition, a perturbation theorem is set up. The global convergence of the algorithm is derived. This result generalizes some existing conclusions. Finally, numerical experiments on two examples demonstrate the effectiveness and efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Many optimal control tasks for engineering problems require the solution of nonconvex optimisation problems, which are rather hard to solve. This article presents a novel iterative optimisation procedure for the fast solution of such problems using successive convexification. The approach considers two types of nonconvexities. Firstly, equality constraints with possibly multiple univariate nonlinearities, which can arise for nonlinear system dynamics. Secondly, nonconvex sets comprised of convex subsets, which may occur for semi-active actuators. By introducing additional decision variables and constraints, the decision variable space is decomposed into affine segments yielding a convex subproblem which is solved efficiently in an iterative manner. Under certain conditions, the algorithm converges to a local optimum of a scalable, piecewise linear approximation of the original problem. Furthermore, the algorithm tolerates infeasible initial guesses. Using a single-mass oscillator application, the procedure is compared with a nonlinear programming algorithm, and the sensitivity regarding initial guesses is analysed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of optimal control problems involving continuous control and state inequality constraints where the control is almost smooth. We first employ the control parametrization technique via approximating the control signal by a piecewise linear function. Then, we develop a time scaling transformation procedure for transforming the approximate problem into an equivalent problem that can be solved readily using conventional methods. On this basis, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. The gradient formulas and convergent properties ensure that the transformed unconstrained optimal parameter selection problems can be solved by existing optimization algorithms or software packages. Finally, an example is solved showing the effectiveness and applicability of the approach proposed. [ABSTRACT FROM AUTHOR]

The concept of generalized augmented Lagrange multiplier was introduced originally for nonlinear programming problems. In this paper we further study this concept in the framework of convex cone constrained optimization problems. In particular we establish the relation of optimal solution sets between the primal problem and augmented Lagrange minimization problem. Some discussions on related concepts such as saddle points and exact penalty are given as well. [ABSTRACT FROM AUTHOR]

We present a general theory of exact penalty functions with vectorial (multidimensional) penalty parameter for optimization problems in infinite dimensional spaces. In comparison with the scalar case, the use of vectorial penalty parameters provides much more flexibility, allows one to adaptively and independently take into account the violation of each constraint during optimization process, and often leads to a better overall performance of an optimization method using an exact penalty function. We obtain sufficient conditions for the local and global exactness of penalty functions with vectorial penalty parameters and study convergence of global exact penalty methods with several different penalty updating strategies. In particular, we present a new algorithmic approach to an analysis of the global exactness of penalty functions, which contains a novel characterisation of the global exactness property in terms of behaviour of sequences generated by certain optimization methods. [ABSTRACT FROM AUTHOR]

In this article, we obtain controls that give the time-optimal trajectories for a Dubins airplane in the presence of fixed and moving obstacles. Using an exact penalty function method, we show the existence of the control variable for a Dubins airplane with given initial and final positions so that the airplane could reach its destination in the shortest time while avoiding fixed and moving obstacles. We demonstrate the effectiveness of our method by three examples. [ABSTRACT FROM AUTHOR]

ObjectivesCurrently, how to plan the safe and efficient movement trajectory of an unmanned surface vehicle (USV) in local waters with multiple known obstacle positions is a research hotspot.MethodsFirst, the obstacle areas are treated with simple and effective circular and convex quadrilateral envelopes, and the obstacle avoidance problem is transformed into the state inequality constraint of a time optimal control problem. The time optimal control problem is then transformed into an optimal parameter selection problem by control parameterization and time scale transformation. Finally, for multiple continuous state inequality constraints caused by multiple obstacles, the exact penalty function method is used to append all state constraints to the cost function. The final form of the problem is suitable for solving any effective optimization technique as a nonlinear optimization problem.ResultsThe numerical simulation results show that the planned trajectory successfully avoids the obstacles in the water and conforms to the motion characteristics of USVs.ConclusionsThe results of this study can provide valuable references for the obstacle avoidance problem in USV trajectory planning.

The bioconversion of 1,3-propanediol from glycerol by Klebsiella pneumoniae can be described by a nonlinear dynamic system. Some work has been done on the identification and optimization of the system, in which the dilution rate of glycerol is considered as a constant. However, the demand of glycerol may vary at different fermentation stages, it is reasonable to view the glycerol metabolic system with dilution rate varying with time. In this paper, we model the glycerol metabolic process as a fourteen-dimensional nonlinear dynamical system, where the dilution rate is considered varying with time. Then an optimal discrete-valued control problem for maximizing the average concentration of 1,3-propanediol in the fermentation process is established. To solve the optimization problem, auxiliary control and an exact penalty function are used to convert this problem into a large-scale parameter optimization problem. For better balancing local and global search ability, a competitive particle swarm algorithm with time-varying control factors is proposed which is proved to be faster and more stable than the traditional competitive particle swarm algorithm. Numerical experiments are conducted to show the rationality, effectiveness and applicability of the method proposed.

Generalized Nash Equilibrium Problems (GNEP) have been attracted by many researchers in the field of game theory, operational research, engineering, economics as well as telecommunication in recent two decades. One of the most important classes of GNEP is a convex GNEP with jointly convex or shared constraints which has been studied extensively. It is considered to be one of the most challenging classes of problems in the field. Moreover, there is a gap in the studies on the GNEP with coupling and shared constraints. The aim of this paper is to investigate the relationship between an exact penalty approach and conjugate duality in convex optimization for the GNEP with coupling and shared constraints. In association with necessary optimality conditions, we obtained the parameterized variational inequality problems. This problem has provided an opportunity to solve many other GNEs. Some numerical results are also presented.

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. [ABSTRACT FROM AUTHOR]

We develop a theoretical framework for constrained optimal control problems. Using an exact penalty method, the differential equation as well as the constraints are handled to obtain an abstract nonsmooth infinite dimensional optimization problem. By tools of nonsmooth analysis theory in infinite dimensional spaces, we investigate the relation between the stationary points of the obtained nondifferentiable optimization problem and those of the constrained optimal control problem. An algorithm to minimize the unconstrained problem combined the penalty function and the projection method is also given. [ABSTRACT FROM AUTHOR]

Abstract 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.

In this paper, a new class of unified penalty functions are derived for the semi-infinite optimization problems, which include many penalty functions as special cases. They are proved to be exact in the sense that under Mangasarian–Fromovitz constraint qualification conditions, a local solution of penalty problem is a corresponding local solution of original problem when the penalty parameter is sufficiently large. Furthermore, global convergence properties are shown under some conditions. The paper is concluded with some numerical examples proving the applicability of our methods to PID controller design and linear-phase FIR digital filter design. [ABSTRACT FROM AUTHOR]

Adjoint methods applied to solve optimal control problems (OCPs) have a restriction that the number of constraints shall be less than that of optimization variables. Otherwise, they are less efficient than the forward methods. This paper proposes an efficient adjoint method to solve OCPs for index-11 differential algebraic systems with continuous-time inequality constraints. The continuous-time inequality constraints are not discretized on time grid but transformed into integrals and penalized in the cost through an exact penalty function. Thus, all the constraints except for box constraints on optimization variables can be removed. Furthermore, a lifted implicit Runge-Kutta (IRK) integrator with adjoint sensitivity propagation is employed to accelerate the function and gradient evaluation procedure. Based on a sensitivity update technique, the number of Newton iterations involved in forward simulation can be reduced to one. Besides this, Lagrange interpolation is applied to approximate the states not on collocation points such that integrals in the penalty function can be evaluated on the same grid for forward simulation. Complexity analysis shows that, for the proposed algorithm, computation involved in the sensitivity propagation is comparable to that of forward one. Numerical simulations on the optimal maneuvering a Delta robot demonstrate that the computational speed of the proposed adjoint algorithm is comparable to that of our previous one, which is based on the lifted IRK integrator and forward sensitivity propagation. [ABSTRACT FROM AUTHOR]

Summary: The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed‐endpoint problems for linear time‐varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free‐endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed‐endpoint and variable‐endpoint problems to equivalent free‐endpoint ones is possible under the assumption that the linearized system is completely controllable, and point out some general properties of nonlinear systems under which a global reduction to equivalent free‐endpoint problems can be achieved. In the case of problems with pointwise state inequality constraints, we prove that such problems for linear time‐varying systems and linear evolution equations with convex state constraints can be reduced to equivalent problems without state constraints, provided one uses the L∞ penalty term, and Slater's condition holds true, while for nonlinear systems a local reduction is possible, if a natural constraint qualification is satisfied. Finally, we show that the exact Lp‐penalization of state constraints with finite p is possible for convex problems, if Lagrange multipliers corresponding to the state constraints belong to Lp′, where p′ is the conjugate exponent of p, and for general nonlinear problems, if the cost functional does not depend on the control inputs explicitly. [ABSTRACT FROM AUTHOR]

Summary: In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact. [ABSTRACT FROM AUTHOR]

New approaches to estimating the coefficients of exact penalty functions for constrained optimization problems are considered. The results of computational experiments using simplified procedures for estimating coefficients in solving some classes of problems are presented. Such approaches are most relevant when applying methods of decomposition by variables (generalized Benders decomposition methods). This allows to overcome the difficulties related to an implicit description of the feasible region of the master problem. [ABSTRACT FROM AUTHOR]

Abstract 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.

The article considers differential inclusion with a given set-valued mapping and initial point. It is required to find a solution of this differential inclusion that minimizes an integral functional. Some classical results about the maximum principle for differential inclusions are obtained using the support and exact penalty functions. This is done for differentiable and for non-differentiable set-valued mappings in phase variables.

In this paper the problem of optimal control of a nonlinear ODE system with given boundary conditions and the integral restriction on control is considered. With the help of the theory of exact penalty functions the original problem is reduced to the problem of unconstrained minimization of a nonsmooth functional. The necessary minimum conditions in terms of hypodifferentials are found. A class of problems for which these conditions are also sufficient is distinguished. On the basis of these conditions the hypodifferential descent method is applied to the considered problem. Under some additional assumptions the hypodifferential descent method converges in a certain sense. [ABSTRACT FROM AUTHOR]