On Fictitious Domain Formulations for Maxwell''s Equations

We consider fictitious domain-Lagrange multiplier formulations for variational problems in the space H(curl: Ω{\bf)} derived from Maxwell's equations. Boundary conditions and the divergence constraint are imposed weakly by using Lagrange multipliers. Both the time dependent and time harmonic formulations of the Maxwell's equations are considered, and we derive well-posed formulations for both cases. The variational problem that arises can be discretized by functions that do not satisfy an a-priori divergence constraint.     

On-line optimization design of sliding mode guidance law with multiple constraints

The design of terminal guidance law with impact angle constraint is required for air-to-ground guided weapons to increase their warhead effect. The variable structure guidance law that consists of diving plane guidance and turning plane guidance equations with impact angle constraint is derived, and the saturation function is introduced into the design of reaching law control to weaken the chattering of the guidance system. The influence of four guidance parameters (i.e., reaching law factor, switching item gain, angle error item factor, and boundary layer thickness) on guidance performance is studied and three typical constraints (i.e., heating rate, normal load factor, and dynamic pressure) are analyzed. An optimization model is established for this problem and the feasibility of on-line optimization on guidance law parameters by the Sequential Quadratic Programming (SQP) algorithm is discussed as well. Simulation results show that the on-line optimization of the derived guidance law not only satisfies specified constraints, but also minimizes the fuel cost during the flying course. Moreover, the optimization process can be completed in a few seconds so that it is suitable for on-board applications.     

OPTIMALITY CONDITIONS AND APPROXIMATE OPTIMALITY CONDITIONS IN LOCALLY LIPSCHITZ VECTOR OPTIMIZATION

Abstract

In this paper, we study constrained locally Lipschitz vector optimization problems in which the objective and constraint spaces are Hilbert spaces, the decision space is a Banach space, the dominating cone and the constraint cone may be with empty interior. Necessary optimality conditions for this type of optimization problems are derived. A sufficient condition for the existence of approximate efficient solutions to a general vector optimization problem is presented. Necessary conditions for approximate efficient solutions to a constrained locally Lipschitz optimization problem is obtained.     

Topology optimization considering the fatigue constraint of variable amplitude load based on the equivalent static load approach

In this research, a new layout optimization method is developed to consider high cycle fatigue constraints which occur due to variable amplitude mechanical loading. Although fatigue is a very important property in terms of safety when designing mechanical components, it has rarely been considered in topology optimization with the lack of concept and the difficulty of sensitivity analysis for fatigue constraints calculated from multiaxial cycle counting. For the topology optimization for fatigue constraint, we use transient stress analysis to extract effective stress cycles and Miner's cumulative damage rule to calculate total damage at every spatial element. Because the calculation of the exact sensitivities of a transient system is complex and time consuming for the topology optimization application, this research proposes to use the pseudo-sensitivities of fatigue constraints calculated by applying equivalent static load approach. In addition, as an aggregated fatigue constraint is very sensitive to the changes in stress value which causes some unstable convergences in optimization process, a new scaling approach of the aggregated fatigue damage constraint is developed. To validate the usefulness of the developed approaches, we solved some benchmark topology optimization problems and found that the present method provides physically appropriate layouts with stable optimization convergence.     

Modeling and Optimization of Interference Limited Wireless Communications

Multiple access interference (MAI) limits the system capacity of wireless communications systems applying code division multiple access (CDMA). Provided that channel knowledge is available in the transmitter, the transmitted signal can be preprocessed by multiuser transmission (MUT) methods. As optimization criterion, the bit error probability/rate (BER) is chosen. On the transmit signal, a limited power constraint is imposed, which is a quadratic function of the transmit signal. The nonlinear optimization problem is modeled in this paper. Unfortunately, the problem is non‐convex. However, with iterative nonlinear optimization methods like SQP, local minima can be found with a performance outperforming other MUT methods. The main remaining challenge are low‐complexity optimization algorithms to allow for a real‐time implementation in highdata rate wireless communications. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of ASP flooding based on dynamic scale IDP with mixed-integer

The alkali/surfactant/polymer (ASP) flooding is a complex distributed parameter system (DPS). In this paper, an optimization model of ASP flooding is developed, which takes net present value (NPV) as the performance index, oil/water seepage continuity equations and adsorption diffusion equations of displacing agents as the governing equations, physicochemical algebraic equations and boundary conditions of displacing agents as the constraint equations. To solve the injection concentration and size of each slug of the model and terminal flooding time, a dynamic scale iterative dynamic programming with mixed-integer (DSMI-IDP) is proposed. The essence of slug size is time, it can only be integer. In DSMI-IDP, the integer truncation is carried out by a proportion method after time normalization which can convert the free time terminal problem to a fixed time terminal problem. A dynamic contraction factor and a principle of adjustment factors are introduced to realize the dynamic scale. To test the algorithm, three examples are solved by DSMI-IDP. At last, the DSMI-IDP is applied to optimize an optimization model of ASP flooding. The solving effect is shown by the comparison with IGA, MIDP and trial and error solutions.     

Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization

Inspired by the successful applications of the stochastic optimization with second order stochastic dominance (SSD) model in portfolio optimization, we study new numerical methods for a general SSD model where the underlying functions are not necessarily linear. Specifically, we penalize the SSD constraints to the objective under Slater’s constraint qualification and then apply the well known stochastic approximation (SA) method and the level function method to solve the penalized problem. Both methods are iterative: the former requires to calculate an approximate subgradient of the objective function of the penalized problem at each iterate while the latter requires to calculate a subgradient. Under some moderate conditions, we show that w.p.1 the sequence of approximated solutions generated by the SA method converges to an optimal solution of the true problem. As for the level function method, the convergence is deterministic and in some cases we are able to estimate the number of iterations for a given precision. Both methods are applied to portfolio optimization problem where the return functions are not necessarily linear and some numerical test results are reported.     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

Robust mean variance optimization problem under Rényi divergence information

In this paper, we consider the robust mean variance optimization problem where the probability distribution of assets’ returns is multivariate normal and the uncertain mean and covariance are controlled by a constraint involving Rényi divergence. We present the closed-form solutions for the robust mean variance optimization problem and find that the choice of order parameter which is related to the Rényi divergence measure will not impact optimal portfolio strategy under the cases that the mean vector and the covariance matrix are uncertain, respectively. Moreover, we obtain the closed-form solution for the robust mean variance optimization problem under the case that the mean vector and the covariance matrix are both uncertain. We illustrate the efficiency of our results with an example.     

On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem

This paper is mainly concerned with the classical KKT reformulation and the primal KKT reformulation (also known as an optimization problem with generalized equation constraint (OPEC)) of the optimistic bilevel optimization problem. A generalization of the MFCQ to an optimization problem with operator constraint is applied to each of these reformulations, hence leading to new constraint qualifications (CQs) for the bilevel optimization problem. M- and S-type stationarity conditions tailored for the problem are derived as well. Considering the close link between the aforementioned reformulations, similarities and relationships between the corresponding CQs and optimality conditions are highlighted. In this paper, a concept of partial calmness known for the optimal value reformulation is also introduced for the primal KKT reformulation and used to recover the M-stationarity conditions.     

Optimization of Algorithmic Parameters using a Meta-Control Approach*

Optimization algorithms usually rely on the setting of parameters, such as barrier coefficients. We have developed a generic meta-control procedure to optimize the behavior of given iterative optimization algorithms. In this procedure, an optimal continuous control problem is defined to compute the parameters of an iterative algorithm as control variables to achieve a desired behavior of the algorithm (e.g., convergence time, memory resources, and quality of solution). The procedure is illustrated with an interior point algorithm to control barrier coefficients for constrained nonlinear optimization. Three numerical examples are included to demonstrate the enhanced performance of this method. This work was primarily done when Z. Zabinsky was visiting Clearsight Systems Inc.     

Multiobjective optimization problem with variational inequality constraints

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem. Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002 Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming problem – Preference – Utility function – Subdifferential calculus – Variational principle Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496 Mathematics Subject Classification (2000): Sub49K24, 90C29     

Convex programming with single separable constraint and bounded variables

In this paper a minimization problem with convex objective function subject to a separable convex inequality constraint “≤” and bounded variables (box constraints) is considered. We propose an iterative algorithm for solving this problem based on line search and convergence of this algorithm is proved. At each iteration, a separable convex programming problem with the same constraint set is solved using Karush-Kuhn-Tucker conditions. Convex minimization problems subject to linear equality/ linear inequality “≥” constraint and bounds on the variables are also considered. Numerical illustration is included in support of theory.     

Static and dynamic VaR constrained portfolios with application to delegated portfolio management

We give a closed-form solution to the single-period portfolio selection problem with a Value-at-Risk (VaR) constraint in the presence of a set of risky assets with multivariate normally distributed returns and the risk-less account, without short sales restrictions. The result allows to obtain a very simple, myopic dynamic portfolio policy in the multiple period version of the problem. We also consider mean-variance portfolios under a probabilistic chance (VaR) constraint and give an explicit solution. We use this solution to calculate explicitly the bonus of a portfolio manager to include a VaR constraint in his/her portfolio optimization, which we refer to as the price of a VaR constraint.     

Parameter estimation for von Mises–Fisher distributions

When analyzing high-dimensional data, it is often appropriate to pay attention only to the direction of each datum, disregarding its norm. The von Mises–Fisher (vMF) distribution is a natural probability distribution for such data. When we estimate the parameters of vMF distributions, parameter κ which corresponds to the degree of concentration is difficult to obtain, and some approximations are necessary. In this article, we propose an iterative algorithm using fixed points to obtain the maximum likelihood estimate (m.l.e.) for κ. We prove that there is a unique local maximum for κ. Besides, using a specific function to calculate the m.l.e., we obtain the upper and lower bounds of the interval in which the exact m.l.e. exists. In addition, based on these bounds, a new and good approximation is derived. The results of numerical experiments demonstrate the new approximation exhibits higher precision than traditional ones.     

A note on the prior parameter choice in finite mixture models of distributions from exponential families

This work presents a new scheme to obtain the prior distribution parameters in the framework of Rufo et al. (Comput Stat 21:621–637, 2006). Firstly, an analytical expression of the proposed Kullback–Leibler divergence is derived for each distribution in the considered family. Therefore, no previous simulation technique is needed to estimate integrals and thus, the error related to this procedure is avoided. Secondly, a global optimization algorithm based on interval arithmetic is applied to obtain the prior parameters from the derived expression. The main advantage by using this approach is that all solutions are found and rightly bounded. Finally, an application comparing this strategy with the previous one illustrates the proposal.     

Manufacturing constraints in parameter–free shape optimization

Structural shape optimization has become an important tool for engineers when it comes to improving components with respect to a given goal function. During this process the designer has to ensure that the optimized part stays manufacturable. Depending on the manufacturing process several requirements could be relevant such as demolding or different kinds of symmetry. This work introduces two approaches on how to handle manufacturing constraints in parameter-free shape optimization. In the so–called explicit approach equality and inequality equations are formulated using the coordinates of the FE-nodes. These equations can be used to extend the optimization problem. Since the number of the additional constraint equations may be very large we apply aggregation formulations, e.g. the Kreisselmeier-Steinhauser function, if necessary. In the second approach, the so–called implicit method, the set of design nodes is split in two groups called optimization nodes and dependent nodes. The optimization nodes are now handled as design nodes but the dependent nodes are coupled to the optimization nodes in such a way that the manufacturing constraint is fulfilled. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An efficient optimization method for mechanical shells considering cut-outs

In this contribution an optimization method for shell structures is presented. This method was developed in order to perform a simultaneous optimization of the shape and position of the mid surface and a topology optimization to introduce cut-outs. A topology optimization method for continuum structures is combined with a manufacturing constraint for deep drawable sheet metals. It is shown, how more than a million design variables can be handled efficiently using a mathematical optimization algorithm for the design update and the finite element method for the structural simulation. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Outer Trust-Region Method for Constrained Optimization

Given an algorithm A for solving some mathematical problem based on the iterative solution of simpler subproblems, an outer trust-region (OTR) modification of A is the result of adding a trust-region constraint to each subproblem. The trust-region size is adaptively updated according to the behavior of crucial variables. The new subproblems should not be more complex than the original ones, and the convergence properties of the OTR algorithm should be the same as those of Algorithm A. In the present work, the OTR approach is exploited in connection with the “greediness phenomenon” of nonlinear programming. Convergence results for an OTR version of an augmented Lagrangian method for nonconvex constrained optimization are proved, and numerical experiments are presented.