Robust penalty function method for an uncertain multi-time control optimization problems

This paper gives some new results on multi-time first-order PDE constrained control optimization problem in the face of data uncertainty (MCOPU). We obtain the robust sufficient optimality conditions for (MCOPU). Further, we construct an unconstrained multi-time control optimization problem (MCOPU)_? corresponding to (MCOPU) via absolute value penalty function method. Then, we show that the robust optimal solution to the constrained problem and a robust minimizer to the unconstrained problem are equivalent under suitable hypotheses. Moreover, we give some non-trivial examples to validate the results established in this paper.     

Convex approximations to sparse PCA via Lagrangian duality

We derive a convex relaxation for cardinality constrained Principal Component Analysis (PCA) by using a simple representation of the L₁ unit ball and standard Lagrangian duality. The resulting convex dual bound is an unconstrained minimization of the sum of two nonsmooth convex functions. Applying a partial smoothing technique reduces the objective to the sum of a smooth and nonsmooth convex function for which an efficient first order algorithm can be applied. Numerical experiments demonstrate its potential.     

A superlinear space decomposition algorithm for constrained nonsmooth convex program

A class of constrained nonsmooth convex optimization problems, that is, piecewise C² convex objectives with smooth convex inequality constraints are transformed into unconstrained nonsmooth convex programs with the help of exact penalty function. The objective functions of these unconstrained programs are particular cases of functions with primal-dual gradient structure which has connection with VU space decomposition. Then a VU space decomposition method for solving this unconstrained program is presented. This method is proved to converge with local superlinear rate under certain assumptions. An illustrative example is given to show how this method works.     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

An approach to constrained global optimization based on exact penalty functions

In the field of global optimization many efforts have been devoted to solve unconstrained global optimization problems. The aim of this paper is to show that unconstrained global optimization methods can be used also for solving constrained optimization problems, by resorting to an exact penalty approach. In particular, we make use of a non-differentiable exact penalty function ${P_q(x;\varepsilon)}$ . We show that, under weak assumptions, there exists a threshold value ${\bar \varepsilon >0 }$ of the penalty parameter ${\varepsilon}$ such that, for any ${\varepsilon \in (0, \bar \varepsilon]}$ , any global minimizer of P _q is a global solution of the related constrained problem and conversely. On these bases, we describe an algorithm that, by combining an unconstrained global minimization technique for minimizing P _q for given values of the penalty parameter ${\varepsilon}$ and an automatic updating of ${\varepsilon}$ that occurs only a finite number of times, produces a sequence {x ^k } such that any limit point of the sequence is a global solution of the related constrained problem. In the algorithm any efficient unconstrained global minimization technique can be used. In particular, we adopt an improved version of the DIRECT algorithm. Some numerical experimentation confirms the effectiveness of the approach.     

L ∞-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

We look at L ^∞-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations. In so doing, use is made of mixed finite element methods. The state and costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control, by piecewise constant functions. L ^∞-error estimates of optimal order are derived for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, numerical tests are presented which confirm our theoretical results.     

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic- parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ Ll-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.     

球约束凸二次规划的一个新算法

首先利用Lagrange对偶 ,将球约束凸二次规划问题转化为无约束优化问题 ,然后运用单纯形法求解无约束优化问题 ,从而获得原问题的最优解     

A Newton‐type algorithm for solving an extremal constrained interpolation problem

Given convex scattered data in R³ we consider the constrained interpolation problem of finding a smooth, minimal L _p‐norm (1 < p < ∞) interpolation network that is convex along the edges of an associated triangulation. In previous work the problem has been reduced to the solution of a nonlinear system of equations. In this paper we formulate and analyse a Newton‐type algorithm for solving the corresponding type of systems. The correctness of the application of the proposed method is proved and its superlinear (in some cases quadratic) convergence is shown. Copyright © 2000 John Wiley & Sons, Ltd.     

Entropic perturbation method for solving a system of linear inequalities

The problem of finding an $\text{x∈}\text{R}{}^{\mathrm{n}}$ such that Ax⩽b and x⩾0 arises in numerous contexts. We propose a new optimization method for solving this feasibility problem. After converting Ax⩽b into a system of equations by introducing a slack variable for each of the linear inequalities, the method imposes an entropy function over both the original and the slack variables as the objective function. The resulting entropy optimization problem is convex and has an unconstrained convex dual. If the system is consistent and has an interior solution, then a closed-form formula converts the dual optimal solution to the primal optimal solution, which is a feasible solution for the original system of linear inequalities. An algorithm based on the Newton method is proposed for solving the unconstrained dual problem. The proposed algorithm enjoys the global convergence property with a quadratic rate of local convergence. However, if the system is inconsistent, the unconstrained dual is shown to be unbounded. Moreover, the same algorithm can detect possible inconsistency of the system. Our numerical examples reveal the insensitivity of the number of iterations to both the size of the problem and the distance between the initial solution and the feasible region. The performance of the proposed algorithm is compared to that of the surrogate constraint algorithm recently developed by Yang and Murty. Our comparison indicates that the proposed method is particularly suitable when the number of constraints is larger than that of the variables and the initial solution is not close to the feasible region.     

Convex approximations in stochastic programming by semidefinite programming

The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in L ₁, L _?? and L ₂ norm. Extensive numerical experiments to investigate the behavior of the proposed methods are also performed.     

函数空间中总极值的R-收敛有限维逼近

应用测度序列R-收敛的新概念来描述函数空间中总极值问题解的有限维逼近,并利用变差积分途径来寻找这样的解.针对有约束问题,运用罚变差积分算法把所给问题转化为无约束问题,且给出一个非凸状态约束最优控制问题的数值例子以说明该算法的有效性.     

有限维逼近无限维总极值的积分型方法

本文用有限维逼近无限维的方法来讨论函数空间中的总体最优化问题．我们给出了新的最优性条件和用变测度方法求得的有限维解逼近总体最优解的算法．对于有约柬问题，我们用不连续罚函数法把有约束问题化为无约束问题来求解．最后，我们通过一个具有非凸状态约束的最优控制问可题的实例来说明算法的有效性．     

Necessary conditions for optimal control problems with bounded state by a penalty method

Necessary conditions are derived for a general relaxed control problem with unilateral state constraint. The results are also valid for ordinary controls that are solutions of the relaxed problem.A penalty is imposed to change the constrained problem into a sequence of unconstrained problems. The assumptions are on the data of the problem and do not requirea priori verification of hypotheses involving the optimal solution.     

An efficient algorithm for facility location in the presence of forbidden regions

This paper investigates a constrained form of the classical Weber problem. Specifically, we consider the problem of locating a new facility in the presence of convex polygonal forbidden regions such that the sum of the weighted distances from the new facility to n existing facilities is minimized. It is assumed that a forbidden region is an area in the plane where travel and facility location are not permitted and that distance is measured using the Euclidean-distance metric. A solution procedure for this nonconvex programming problem is presented. It is shown that by iteratively solving a series of unconstrained problems, this procedure terminates at a local optimum to the original constrained problem. Numerical examples are presented.     

Sufficient second-order optimality conditions for convex control constraints

In this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints. Here, the control is a function with values in Rn. The constraint is of the form u(x)∈U(x), where U is a set-valued mapping that is assumed to be measurable with convex and closed images. The second-order condition requires coercivity of the Lagrange function on a suitable subspace, which excludes strongly active constraints, together with first-order necessary conditions. It ensures local optimality of a reference function in an L^∞-neighborhood. The analysis is done for a model problem namely the optimal distributed control of the instationary Navier-Stokes equations.     

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Summary In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ¹ discretized by using piecewise quadratic finite elements and in ² by using the five-point difference approximation are presented.     

Optimal control of a system governed by hyperbolic operator with an infinite number of variables

A distributed control problem for the hyperbolic operator with an infinite number of variables is considered. The controls are allowed to be in the space L₂(0, T; L₂(R^∞)) = L₂(Q). The necessary and sufficient condition for the control to be optimal is obtained and the set of inequalities that characterize this condition is also obtained.     

A coordinate gradient descent method for nonsmooth separable minimization

We consider the problem of minimizing the sum of a smooth function and a separable convex function. This problem includes as special cases bound-constrained optimization and smooth optimization with ?₁-regularization. We propose a (block) coordinate gradient descent method for solving this class of nonsmooth separable problems. We establish global convergence and, under a local Lipschitzian error bound assumption, linear convergence for this method. The local Lipschitzian error bound holds under assumptions analogous to those for constrained smooth optimization, e.g., the convex function is polyhedral and the smooth function is (nonconvex) quadratic or is the composition of a strongly convex function with a linear mapping. We report numerical experience with solving the ?₁-regularization of unconstrained optimization problems from Moré et al. in ACM Trans. Math. Softw. 7, 17–41, 1981 and from the CUTEr set (Gould and Orban in ACM Trans. Math. Softw. 29, 373–394, 2003). Comparison with L-BFGS-B and MINOS, applied to a reformulation of the ?₁-regularized problem as a bound-constrained optimization problem, is also reported.     

Variational perturbations of the linear-quadratic problem

A sequence of perturbations $$\begin{gathered} \dot x = A_n x + B_n u, \parallel u\parallel _{L^2 } \leqslant 1, (P_n ) \hfill \\ x\left( o \right) = x^o , n = 0, 1, 2, 3,..., \hfill \\ \end{gathered} $$ is given of the linear-quadratic optimal control problem consisting of minimizing $$\int_0^1 {((u - \tilde u)^T (u - \tilde u) + (x - \tilde x)^T (x - \tilde x))dt,} $$ subject to (P₀). We assume that {A _n} bounded inL ¹ and {B _n} is bounded inL ². Then, a necessary and sufficient condition so that, for every?, \(\tilde u\) , \(\tilde x\) ∈L ², and for everyx ⁰, the optimal control for (P_n) converges strongly inL ² to the optimal control for (P₀) and the optimal state converges uniformly is thatA _n →A ₀ weakly inL ¹ andB _n →B ₀ strongly inL ².