A robust algorithm for generalized geometric programming

Most existing methods of global optimization for generalized geometric programming (GGP) actually compute an approximate optimal solution of a linear or convex relaxation of the original problem. However, these approaches may sometimes provide an infeasible solution, or far from the true optimum. To overcome these limitations, a robust solution algorithm is proposed for global optimization of (GGP) problem. This algorithm guarantees adequately to obtain a robust optimal solution, which is feasible and close to the actual optimal solution, and is also stable under small perturbations of the constraints.     

Performance analysis of distributed solution approaches in simulation-based optimization

Applying computationally expensive simulations in design or process optimization results in long-running solution processes even when using a state-of-the-art distributed algorithm and hardware. Within these simulation-based optimization problems the optimizer has to treat the simulation systems as black-boxes. The distributed solution of this kind of optimization problem demands efficient utilization of resources (i.e. processors) and evaluation of the solution quality. Analyzing the parallel performance is therefore an important task in the development of adequate distributed approaches taking into account the numerical algorithm, its implementation, and the used hardware architecture. In this paper, simulation-based optimization problems are characterized and a distributed solution algorithm is presented. Different performance analysis techniques (e.g. scalability analysis, computational complexity) are discussed and a new approach integrating parallel performance and solution quality is developed. This approach combines a priori and a posteriori techniques and can be applied in early stages of the solution process. The feasibility of the approach is demonstrated by applying it to three different classes of simulation-based optimization problems from groundwater management.     

A robust algorithm for quadratic optimization under quadratic constraints

Most existing methods of quadratically constrained quadratic optimization actually solve a refined linear or convex relaxation of the original problem. It turned out, however, that such an approach may sometimes provide an infeasible solution which cannot be accepted as an approximate optimal solution in any reasonable sense. To overcome these limitations a new approach is proposed that guarantees a more appropriate approximate optimal solution which is also stable under small perturbations of the constraints.     

An option-based revenue management procedure for strategic airline alliances

An airline has to decide whether to accept an incoming customer request for a seat in the airplane or to reject it in hope that another customer will request the seat later at a higher price. Capacity control, as one of the instruments of revenue management, gives a solution to this decision problem. In the presence of strategic alliances capacity control changes. For the case of two airlines in the alliance and a single flight leg we propose an option-based capacity control process. The determination of booking limits for capacity control is done with real options. A simulation model is introduced to evaluate the booking process of the partner airlines within the strategic alliance, considering the option-based procedure. In an iterative process the booking limits are improved with simulation-based optimization. The results of the option-based procedure will be compared with the results of the simulation-based optimization, the results of a first-come-first-served (FCFS) approach and ex post optimal solutions.     

On Smoothing l1 Exact Penalty Function for Constrained Optimization Problems

This article introduces a smoothing technique to the l₁ exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.     

Approximate solution of the trust region problem by minimization over two-dimensional subspaces

The trust region problem, minimization of a quadratic function subject to a spherical trust region constraint, occurs in many optimization algorithms. In a previous paper, the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two-dimensional trust region problem. They showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem. This paper reports computational results showing that the two-dimensional minimization approach gives nearly optimal reductions in then-dimension quadratic model over a wide range of test cases. We also show that there is very little difference, in efficiency and reliability, between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization. These results may encourage the application of similar approximate trust region techniques in other contexts.Research supported by ARO contract DAAG 29-84-K-0140, NSF grant DCR-8403483, and NSF cooperative agreement DCR-8420944.     

Bayesian portfolio selection with multi-variate random variance models

We consider multi-period portfolio selection problems for a decision maker with a specified utility function when the variance of security returns is described by a discrete time stochastic model. The solution of these problems involves a dynamic programming formulation and backward induction. We present a simulation-based method to solve these problems adopting an approach which replaces the preposterior analysis by a surface fitting based optimization approach. We provide examples to illustrate the implementation of our approach.     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.     

AN OPTIMIZATION APPROACH TO IDENTIFYING TIMBER SUPPLY FUNCTION COEFFICIENTS

This paper examines an optimization approach to identifying short-run timber supply function coefficients when the form of the supply function is known. By definition, a short-run timber supply function is a functional relationship between the optimal harvest level in each period (e.g., each year) and the actual forest-market state in the same period. The short-run timber supply function represents the optimal harvest decision policy, and therefore, the problem of optimal harvesting can be formulated as a problem of determining this function. When the form of the supply function is known, the problem becomes one of identifying the coefficients of the supply function. If the management objective is to maximize the expected present value of net revenues from timber harvesting over an infinite time horizon, and the timber price process is, in a sense, stationary, the supply function coefficients correspond to the optimal solution to an anticipative optimization problem. In this case, the supply function coefficients can be determined by maximizing the expected present value of the net revenues from timber harvesting, where periodic harvest levels are determined using the supply function. Numerical results show that the short-run supply functions determined using this approach gives good approximations of the true supply function.     

$${{\mathcal {D}(\mathcal {C})}}$$-optimization and robust global optimization

For solving global optimization problems with nonconvex feasible sets existing methods compute an approximate optimal solution which is not guaranteed to be close, within a given tolerance, to the actual optimal solution, nor even to be feasible. To overcome these limitations, a robust solution approach is proposed that can be applied to a wide class of problems called D(C){{\mathcal {D}(\mathcal {C})}}-optimization problems. DC optimization and monotonic optimization are particular cases of D(C){{\mathcal {D}(\mathcal {C})}}-optimization, so this class includes virtually every nonconvex global optimization problem of interest. The approach is a refinement and extension of an earlier version proposed for dc and monotonic optimization.     

Approximating Pareto curves using semidefinite relaxations

We approximate as closely as desired the Pareto curve associated with bicriteria polynomial optimization problems. We use three formulations (including the weighted sum approach and the Chebyshev approximation) and each of them is viewed as a parametric polynomial optimization problem. For each case is associated a hierarchy of semidefinite relaxations and from an optimal solution of each relaxation one approximates the Pareto curve by solving an inverse problem (first two cases) or by building a polynomial underestimator (third case).     

A derivative-free approach for a simulation-based optimization problem in healthcare

Hospitals have been challenged in recent years to deliver high quality care with limited resources. Given the pressure to contain costs, developing procedures for optimal resource allocation becomes more and more critical in this context. Indeed, under/overutilization of emergency room and ward resources can either compromise a hospital’s ability to provide the best possible care, or result in precious funding going toward underutilized resources. Simulation-based optimization tools then help facilitating the planning and management of hospital services, by maximizing/minimizing some specific indices (e.g. net profit) subject to given clinical and economical constraints. In this work, we develop a simulation-based optimization approach for the resource planning of a specific hospital ward. At each step, we first consider a suitably chosen resource setting and evaluate both efficiency and satisfaction of the restrictions by means of a discrete-event simulation model. Then, taking into account the information obtained by the simulation process, we use a derivative-free optimization algorithm to modify the given setting. We report results for a real-world problem coming from the obstetrics ward of an Italian hospital showing both the effectiveness and the efficiency of the proposed approach.     

Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem

Optimization is of vital importance when performing intensity modulated radiation therapy to treat cancer tumors. The optimization problem is typically large-scale with a nonlinear objective function and bounds on the variables, and we solve it using a quasi-Newton sequential quadratic programming method. This study investigates the effect on the optimal solution, and hence treatment outcome, when solving an approximate optimization problem of lower dimension. Through a spectral decompostion, eigenvectors and eigenvalues of an approximation to the Hessian are computed. An approximate optimization problem of reduced dimension is formulated by introducing eigenvector weights as optimization parameters, where only eigenvectors corresponding to large eigenvalues are included. The approach is evaluated on a clinical prostate case. Compared to bixel weight optimization, eigenvector weight optimization with few parameters results in faster initial decline in the objective function, but with inferior final solution. Another approach, which combines eigenvector weights and bixel weights as variables, gives lower final objective values than what bixel weight optimization does. However, this advantage comes at the expense of the pre-computational time for the spectral decomposition. A preliminary version of this paper was presented at the AAPM 46th annual meeting, held July 25–29, 2004 in Pittsburgh, PA.     

An approach for solving of a moving boundary problem

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.     

The exponential matrix function and linear differential equations

Monte Carlo optimization has been shown to be useful in solving multivariate optimization problems. In most Monte Carlo simulations, to find the optimal solution of an integer programming problem, the computer gets close to the true optimal solution, but does not find the exact optimal. This can be a problem at times but it can be overcome by multi‐stage Monte Carlo integer programs that find a preliminary nearly optimal solution and use this as a focal point and springboard for finding the true optimal.     

Robust Solution of Nonconvex Global Optimization Problems

The concept of ɛ-approximate optimal solution as widely used in nonconvex global optimization is not quite adequate, because such a point may correspond to an objective function value far from the true optimal value, while being infeasible. We introduce a concept of essential ɛ-optimal solution, which gives a more appropriate approximate optimal solution, while being stable under small perturbations of the constraints. A general method for finding an essential ɛ-optimal solution in finitely many steps is proposed which can be applied to d.c. programming and monotonic optimization.     

Progressive genetic algorithm for solution of optimization problems with nonlinear equality and inequality constraints

A new approach, identified as progressive genetic algorithm (PGA), is proposed for the solutions of optimization problems with nonlinear equality and inequality constraints. Based on genetic algorithms (GAs) and iteration method, PGA divides the optimization process into two steps; iteration and search steps. In the iteration step, the constraints of the original problem are linearized using truncated Taylor series expansion, yielding an approximate problem with linearized constraints. In the search step, GA is applied to the problem with linearized constraints for the local optimal solution. The final solution is obtained from a progressive iterative process. Application of the proposed method to two simple examples is given to demonstrate the algorithm.     

Optimal operational service levels in vendor managed inventory contracts - an exact approach

Vendor Managed Inventory (VMI) contracts are anchored on a fill rate at which the vendor is expected to meet the end-customer demand. Violations of this contracted fill rate due to excess and insufficient inventory are both penalized, often in a linear, but asymmetric manner. To minimize these costs, the vendor needs to maintain an operational fill rate that is different from the contracted fill rate. We model, analyze and solve an optimization problem that determines this operational fill rate and the associated optimal inventory decision. We establish that, for some special, yet popular, models of demand (e.g. truncated normal, gamma, Weibull and uniform distributions), the optimal solution can be derived in closed form and computed precisely. For other demand distributions, either the optimization problem becomes ill-defined or we may need to use approximate solution methods. An extensive computational study reveals that, for realistic values of problem parameters, the operational fill rate is often larger (by as much as 20%) than the contracted service level, possibly explaining the inventory glut commonly observed in real-world VMI systems.     

A realization of constraint feasibility in a moving least squares response surface based approximate optimization

In the context of approximate optimization, the most extensively used tools are the response surface method (RSM) and the moving least squares method (MLSM). Since traditional RSMs and MLSMs are generally described by second-order polynomials, approximate optimal solutions can, at times, be infeasible in cases where highly nonlinear and/or nonconvex constraint functions are to be approximated. This paper explores the development of a new MLSM-based meta-model that ensures the constraint feasibility of an approximate optimal solution. A constraint-feasible MLSM, referred to as CF-MLSM, makes approximate optimization possible for all of the convergence processes, regardless of the multimodality/nonlinearity in the constraint function. The usefulness of the proposed approach is verified by examining various nonlinear function optimization problems.