Infeasible Point Research Articles

ON OPTIMUM DESIGN OF FRAME STRUCTURES

Optimization of frame structures is formulated as a non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefinite programming, and (iv) polynomial optimization. We show that polynomial optimization solves the frame structure optimization to global optimality by building the (moment-sums-of-squares) hierarchy of convex linear semidefinite programming problems, and it also provides guaranteed lower and upper bounds on optimal design. Finally, we solve three sample optimization problems and conclude that the local optimization approaches may indeed converge to local optima, without any solution quality measure, or even to infeasible points. These issues are readily overcome by using polynomial optimization, which exhibits a finite convergence, at the prize of higher computational demands.

Multi-objective path planning of an autonomous mobile robot using hybrid PSO-MFB optimization algorithm

The main aim of this paper is to solve a path planning problem for an autonomous mobile robot in static and dynamic environments. The problem is solved by determining the collision-free path that satisfies the chosen criteria for shortest distance and path smoothness. The proposed path planning algorithm mimics the real world by adding the actual size of the mobile robot to that of the obstacles and formulating the problem as a moving point in the free-space. The proposed algorithm consists of three modules. The first module forms an optimized path by conducting a hybridized Particle Swarm Optimization-Modified Frequency Bat (PSO-MFB) algorithm that minimizes distance and follows path smoothness criteria. The second module detects any infeasible points generated by the proposed hybrid PSO-MFB Algorithm by a novel Local Search (LS) algorithm integrated with the hybrid PSO-MFB algorithm to be converted into feasible solutions. The third module features obstacle detection and avoidance (ODA), which is triggered when the mobile robot detects obstacles within its sensing region, allowing it to avoid collision with obstacles. The simulation results indicate that this method generates an optimal feasible path even in complex dynamic environments and thus overcomes the shortcomings of conventional approaches such as grid methods. Moreover, compared to recent path planning techniques, simulation results show that the proposed hybrid PSO-MFB algorithm is highly competitive in terms of path optimality.

Repairing constrained experimental regions and designs for mixture or nonmixture variables when some design points are unacceptable

A constrained experimental region (CER) involving mixture variables or nonmixture variables is generally specified by lower and/or upper bounds on the variables and possibly by lower and/or upper bounds on linear combinations of the variables. Additionally, the proportions of mixture variables typically are constrained to sum to unity. In some cases, an experimental design generated to explore the CER may contain unacceptable points (e.g., infeasible points or points with undesirable response values). This may not be discovered until after the experiment has been completed. In such cases, it can be desirable to repair the experimental design and the CER so that they contain only acceptable points. Two approaches for performing such repairs are discussed. The first approach repairs the CER to exclude unacceptable subregions, then uses existing optimal design methodology to select design points to repair the design. The new second approach (i) replaces each unacceptable design point with an acceptable design point and (ii) adds one new constraint associated with each replacement design point to repair the CER. Both approaches are illustrated with a three-component mixture experiment example that permits visualization. A 15-component mixture experiment example is also used to illustrate the new second approach, which is reasonable for experiments with many variables. Finally, an example involving three nonmixture variables is provided in supplementary material. Applying the approaches for repairing experimental designs and CERs to problems with both mixture and nonmixture variables is briefly discussed.

Space-filling experimental designs for constrained design spaces

ABSTRACTConventional space-filling experimental design provides uniform coverage of a hypercube design space. When constraints are imposed, the results may contain many infeasible points. Simply omitting these points leads to fewer feasible points than desired and a design of experiments that is not optimally distributed. In this research, an adaptive method is developed to create space-filling points in arbitrarily constrained spaces. First, a design space reconstruction method is developed to reduce the invalid exploration space and enhance the efficiency of experimental designs. Then, a synthetic criterion of uniformity and feasibility is proposed and optimized by the enhanced stochastic evolutionary method to obtain the initial sampling combination. Finally, an adaptive adjustment strategy of design levels is constructed to obtain the required number of feasible points. Various test cases with convex and non-convex, connected and non-connected design spaces are implemented to verify the efficacy of the proposed method.

Identifying and Characterizing Non-Convexities in Feasible Spaces of Optimal Power Flow Problems

Optimal power flow (OPF) is an important problem in the operation of electric power systems. The solution to an OPF problem provides a minimum cost operating point that satisfies constraints imposed by both the non-linear power flow equations and engineering limits. These constraints can yield non-convex feasible spaces that result in significant computational challenges. This brief proposes an algorithm that identifies and characterizes non-convexities in OPF feasible spaces. This algorithm searches for a pair of feasible points whose connecting line segment contains an infeasible point. Such points certify the existence of a non-convexity in the OPF feasible space. Moreover, the constraint violations at the infeasible point along the connecting line segment physically characterize a cause of the non-convexity. Numerical demonstrations include a small illustrative example as well as applications to various test cases.

Convergence-order analysis of branch-and-bound algorithms for constrained problems

The performance of branch-and-bound algorithms for deterministic global optimization is strongly dependent on the ability to construct tight and rapidly convergent schemes of lower bounds. One metric of the efficiency of a branch-and-bound algorithm is the convergence order of its bounding scheme. This article develops a notion of convergence order for lower bounding schemes for constrained problems, and defines the convergence order of convex relaxation-based and Lagrangian dual-based lower bounding schemes. It is shown that full-space convex relaxation-based lower bounding schemes can achieve first-order convergence under mild assumptions. Furthermore, such schemes can achieve second-order convergence at KKT points, at Slater points, and at infeasible points when second-order pointwise convergent schemes of relaxations are used. Lagrangian dual-based full-space lower bounding schemes are shown to have at least as high a convergence order as convex relaxation-based full-space lower bounding schemes. Additionally, it is shown that Lagrangian dual-based full-space lower bounding schemes achieve first-order convergence even when the dual problem is not solved to optimality. The convergence order of some widely-applicable reduced-space lower bounding schemes is also analyzed, and it is shown that such schemes can achieve first-order convergence under suitable assumptions. Furthermore, such schemes can achieve second-order convergence at KKT points, at unconstrained points in the reduced-space, and at infeasible points under suitable assumptions when the problem exhibits a specific separable structure. The importance of constraint propagation techniques in boosting the convergence order of reduced-space lower bounding schemes (and helping mitigate clustering in the process) for problems which do not possess such a structure is demonstrated.

Assessing the reliability of general-purpose Inexact Restoration methods

Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper [Banihashemi and Kaya (2013)] suggests that the Inexact Restoration approach can be competitive with well-established nonlinear programming solvers when applied to certain control problems without any problem-oriented procedure for restoring feasibility. This result motivated us to revisit the idea of designing general-purpose Inexact Restoration methods, especially for large-scale problems. In this paper we introduce affordable algorithms of Inexact Restoration type for solving arbitrary nonlinear programming problems and we perform the first experiments that aim to assess their reliability. Initially, we define a purely local Inexact Restoration algorithm with quadratic convergence. Then, we modify the local algorithm in order to increase the chances of success of both the restoration and the optimization phase. This hybrid algorithm is intermediate between the local algorithm and a globally convergent one for which, under suitable assumptions, convergence to KKT points can be proved.

Extended Modulation Index with more Eliminated Low Order Harmonics at Cascaded Multilevel Inverters

Principle aim in standard Selective Harmonic Elimination pulse wide modulation (SHE-PWM) is elimination of low-order harmonics as well as production of desired fundamental component. To accomplish this objective, a set of N switching angles should be determined. In fact, N-1 angles are hired to remove N-1 harmonics from voltage waveform and one angle is allotted to produce the fundamental. This paper presents a solution in order to remove low order harmonics as much as possible. It suggests eliminating next harmonic instead of highest order harmonic in infeasible points. A genetic algorithm is employed to solve the new transcendental equation for finding the switching angles. The proposed method is experimentally applied to a seven-level cascaded H-bridge inverter. The results confirm the validity and effectiveness of the proposed method in extending modulation index range.

Performance Entitlement of Supercritical Steam Bottoming Cycle

A supercritical steam bottoming cycle has been proposed as a performance enhancement option for gas turbine combined cycle power plants. The technology has been widely used in coal-fired steam turbine power plants since the 1950s and can be considered a mature technology. Its application to the gas-fired combined cycle systems presents unique design challenges due to the much lower gas temperatures (i.e., 650 °C at the gas turbine exhaust vis-à-vis 2000 °C in fossil fuel-fired steam boilers). Thus, the potential impact of the supercritical steam conditions is hampered to the point of economic infeasibility. This technical brief draws upon the second-law based exergy concept to rigorously quantify the performance entitlement of a supercritical high-pressure boiler section in a heat recovery steam generator utilizing the exhaust of a gas turbine to generate steam for power generation in a steam turbine.

Distance confined path problem and separable integer programming

We consider in this article a problem of finding paths in a graph within a distance window, termed distance-confined path problem and develop a successive graph reduction scheme to solve such a problem effectively. Integrated with the surrogate constraint formulation, our proposed solution algorithm for the distance-confined path problem finds a prominent application in separable non-linear integer programming. By forming a family of distance-confined path problems, we are able to remove gradually all primal infeasible points in the region defined by the surrogate constraint and eventually achieve an exact equivalence between the surrogate constraint formulation and the primal problem, thus a zero duality gap.

A tighter cut generation strategy for acceleration of Benders decomposition

This paper presents a novel strategy for speeding up the classical Benders decomposition for large-scale mixed integer linear programming problems. The proposed method is particularly useful when the optimality cut is difficult to obtain. A ratio of distances from a feasible point to an infeasible point and a feasibility cut is used as a metric to determine the tightest constraint for the region located by the feasible point, thus improving the convergence rate. Application of the proposed approach to a multi-product batch plant scheduling problem shows substantial improvement both in the computational time and the number of iterations.

A constrained optimization algorithm based on the simplex search method

In this article, a robust method is presented for handling constraints with the Nelder and Mead simplex search method, which is a direct search algorithm for multidimensional unconstrained optimization. The proposed method is free from the limitations of previous attempts that demand the initial simplex to be feasible or a projection of infeasible points to the nonlinear constraint boundaries. The method is tested on several benchmark problems and the results are compared with various evolutionary algorithms available in the literature. The proposed method is found to be competitive with respect to the existing algorithms in terms of effectiveness and efficiency.

Handling infeasibility in a large-scale nonlinear optimization algorithm

Practical Nonlinear Programming algorithms may converge to infeasible points. It is sensible to detect this situation as quickly as possible, in order to have time to change initial approximations and parameters, with the aim of obtaining convergence to acceptable solutions in further runs. In this paper, a recently introduced Augmented Lagrangian algorithm is modified in such a way that the probability of quick detection of asymptotic infeasibility is enhanced. The modified algorithm preserves the property of convergence to stationary points of the sum of squares of infeasibilities without harming the convergence to KKT points in feasible cases.

Improved constraint consensus methods for seeking feasibility in nonlinear programs

The Constraint Consensus method moves quickly from an initial infeasible point to a point that is close to feasibility for a set of nonlinear constraints. It is a useful first step prior to launching an expensive local solver, improving the probability that the local solver will find a solution and the speed with which it is found. The two main ingredients are the method for calculating the feasibility vector for each violated constraint (the estimated vector to the closest point that satisfies the constraint), and the method of combining the feasibility vectors into a single consensus vector that updates the current point. We propose several improvements: (i) a simple new method for calculating the consensus vector, (ii) a predictor-corrector approach to adjusting the consensus vector, (iii) an improved way of selecting the output point, and (iv) ways of selecting subsets of the constraints to operate on at a given iteration. These techniques greatly improve the performance of barrier method local solvers. Quadratic feasibility vectors are also investigated. Empirical results are given for a large set of nonlinear and nonconvex models.

Efficient parametric analysis of the chemical master equation through model order reduction

BackgroundStochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation.ResultsIn this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations.ConclusionsThe results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.

Formulation of the Audze–Eglais uniform Latin hypercube design of experiments for constrained design spaces

Optimal Latin Hypercubes (OLH) created in a constrained design space might produce Design of Experiments (DoE) containing infeasible points if the underlying formulation disregards the constraints. Simply omitting these infeasible points leads to a DoE with fewer experiments than desired and to a set of points that is not optimally distributed. By using the same number of points a better mapping of the feasible space can be achieved. This paper describes the development of a procedure that creates OLHs for constrained design spaces. An existing formulation is extended to meet this requirement. Here, the OLH is found by minimizing the Audze-Eglais potential energy of the points using a permutation genetic algorithm. Examples validate the procedure and demonstrate its capabilities in finding space-filling Latin Hypercubes in arbitrarily shaped design spaces.

Two new predictor-corrector algorithms for second-order cone programming

Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming (SOCP) are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton (AHO) directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O(rln(ɛ0/ɛ)) is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.

Elimination of Low Order Harmonics in Multilevel Inverters Using Genetic Algorithm

The selective harmonic elimination pulse width modulation (SHEPWM) switching strategy has been applied to multilevel inverters to remove low order harmonics. Naturally, the related equations do not have feasible solutions for some operating points associated with the modulation index (M). However, with these infeasible points, minimizing instead of eliminating harmonics is performed. Thus, harmful harmonics such as the 5 th harmonic still remains in the output waveform. Therefore, it is proposed in this paper to ignore solving the equation associated with the highest order harmonics. A reduction in the eliminated harmonics results in an increase in the degrees of freedom. As a result, the lower order harmonics are eliminated in more operating points. A 9-level inverter is chosen as a case study. The genetic algorithm (GA) for optimization purposes is used. Simulation results verify the proposed method.

Mobile element assisted cooperative localization for wireless sensor networks with obstacles

In this paper, a cooperative localization algorithm is proposed that considers the existence of obstacles in mobility-assisted wireless sensor networks (WSNs). An optimal movement scheduling method with mobile elements (MEs) is proposed to address limitations of static WSNs in node localization. In this scheme, a mobile anchor node cooperates with static sensor nodes and moves actively to refine location performance. It takes advantage of cooperation between MEs and static sensors while, at the same time, taking into account the relay node availability to make the best use of beacon signals. For achieving high localization accuracy and coverage, a novel convex position estimation algorithm is proposed, which can effectively solve the problem when infeasible points occur because of the effects of radio irregularity and obstacles. This method is the only rangefree based convex method to solve the localization problem when the feasible set of localization inequalities is empty. Simulation results demonstrate the effectiveness of this algorithm.

Sequential feasible optimal power flow: theoretical basis and numerical implementation

Sequential feasible optimal (SFO) power flow is developed for power systems. One distinguished feature of this algorithm is that it can maintain feasibility for both equality and inequality constraints during iterations. SFO consists of two phases: Phase I is used to obtain a feasible point from any infeasible initial point. The feasible point is obtained by solving an active-set quotient gradient system (AS-QGS) using pseudo-transient continuation method. Starting from the feasible point obtained in Phase I, Phase II yields a series of feasible points until it converges to KKT point. At each iteration of Phase II, the movement from one feasible point to another feasible point with a smaller objective value is achieved by two stages: objective improving stage and feasibility enforcing stage. Numerical studies on several testing systems indicate that the proposed feasible approach can be terminated at any iteration and yield a feasible operating point. Moreover, SFO can also be used to help operators to diagnose the divergence cases, identify the unreasonable constraints. Based on the diagnosis results, operator can adjust it to obtain a convergence case. Copyright © 2009 John Wiley & Sons, Ltd.