Polynomial Objective Function Research Articles

Inhomogeneous polynomial optimization over a convex set: An approximation approach

In this paper, we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set. The focus is on the design and analysis of polynomial-time approximation algorithms. The methods are able to deal with optimization models with polynomial objective functions in any fixed degrees. In particular, we first study the problem of maximizing an inhomogeneous polynomial over the Euclidean ball. A polynomial-time approximation algorithm is proposed for this problem with an assured (relative) worst-case performance ratio, which is dependent only on the dimensions of the model. The method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co-centered ellipsoids. Furthermore, the constraint set is extended to a general convex compact set. Specifically, we propose a polynomial-time approximation algorithm with a (relative) worst-case performance ratio for polynomial optimization over some convex compact sets, e.g. a polytope. Finally, numerical results are reported, revealing good practical performance of the proposed algorithms for solving some randomly generated instances.

Atomic optimization. II. Multidimensional problems and polynomial matrix inequalities

The paper studies multidimensional optimization problems with a polynomial objective function and constraints in the form of polynomial matrix inequalities. The author suggests a transformation of the solution method based on the theory of moments. This transformation allows to reduce appreciably the computational complexity of the method, still preserving its applicability to optimization problems of the above class.

Atomic optimization. I. Search Space Transformation and One-dimensional Problems

The paper studies optimization problems with polynomial objective functions and constraints in the form of inequalities. We propose a transformation of the solution method based on the theory of moments. This transformation allows to construct equivalent solution algorithms in the augmented original search space instead of the space of moments. Onedimensional optimization problems are explored in detail.

Concentration inequalities for nonlinear matroid intersection

In this work we propose new randomized rounding algorithms for matroid intersection and matroid base polytopes. We prove concentration inequalities for polynomial objective functions and constraints that has numerous applications and can be used in approximation algorithms for Minimum Quadratic Spanning Tree, Unrelated Parallel Machines Scheduling and scheduling with time windows and nonlinear objectives. We also show applications related to Constraint Satisfaction and dense polynomial optimization. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 541-571, 2015

The domain of attraction for the endemic equilibrium of an SIRS epidemic model

Abstract In this paper, a new method is adopted to construct a Lyapunov function for the endemic equilibrium of the J. Mena-Lorca and H.W. Hothcote’s SIRS epidemic model with bilinear incidence and constant recruitment. On the basis of the Lyapunov function, the domain of the attraction of the endemic equilibrium is estimated by solving an LMI optimization problem with multivariate polynomial objective function and constraints.

Integer Programming Subject to Monomial Constraints

We investigate integer programs containing constraints of the type $\prod_{i\in I}x_i^{\alpha_i}=b$. Due to the number-theoretic nature of these constraints, standard methods based on linear algebra cannot be applied directly. Instead, we present a reformulation resulting in integer programs with linear constraints and polynomial objective functions, using prime decompositions of the right-hand sides b. Moreover, we show that minimizing a linear objective function with nonnegative coefficients over bivariate constraints is possible in polynomial time.

Analog circuit design by nonconvex polynomial optimization: Two design examples

AbstractWe present a framework for synthesizing low‐power analog circuits through global optimization over generally nonconvex multivariate polynomial objective function and constraints. Specifically, a nonconvex optimization problem is formed, which is then efficiently solved through convex programming techniques based on linear matrix inequality (LMI) relaxation. The framework allows both polynomial inequality and equality constraints, thereby facilitating more accurate device modelings and parameter tuning. Compared to traditional nonlinear programming (NLP), the proposed methodology exhibits superior computational efficiency, and guarantees convergence to a globally optimal solution. As in other physical design tasks, circuit knowledge and insight are critical for initial problem formulation, while the nonconvex optimization machinery provides a versatile tool and systematic way to locate the optimal parameters meeting design specifications. Two circuit design examples are given, namely, a nested transconductance(Gm)–capacitance compensation (NGCC) amplifier and a delta–sigma (ΔΣ) analog‐to‐digital converter (ADC), both of them being the key components in many electronic systems. Copyright © 2008 John Wiley & Sons, Ltd.

Stacking sequence optimization to maximize the buckling load of blade-stiffened panels with strength constraints using the iterative fractal branch and bound method

Laminated carbon fiber reinforced polymer (CFRP) composites have widespread applications in aerospace structures, and thus optimization of the stacking sequences in these composites is indispensable. Here, a fractal branch and bound method (FBB) is proposed for optimizing the stacking sequences. This method requires only low computational costs, and an optimal result can be obtained rapidly by means of the deterministic process. For practical stacking sequence optimizations, more than two laminates have to be optimized, because a practical aerospace structural component usually comprises a panel and stiffeners made from composite laminates. Since the stacking sequences of the skin panel and stiffeners affect the buckling load of the stiffened panel, the optimization of both laminates must be performed simultaneously. In the present study, a new method to implement a strength constraint for the FBB method is proposed for the simultaneous optimization of more than two laminates (such as a panel and stiffeners). Moreover, a quadratic polynomial objective function, which includes lamination parameter variables of the two laminates: the stiffeners and the panel, is adopted. The strength constraint is implemented by means of a response surface. The new method is applied to the buckling load maximization of a blade-stiffened composite panel, in which the strength constraint is demonstrated as a feasibility study. The method successfully obtained optimal stacking sequences with the strength constraint at low computational cost.

AN ANALYTICAL SOLUTION TO MULTIVARIABLE NONLINEAR MPC FOR SECOND-ORDER LAGUERRE SYSTEMS

An analytical solution to the multivariable (two-input, two-output) nonlinear model predictive control (NMPC) problem was derived for systems represented by (Volterra-)Laguerre models. The standard two-norm squared NMPC objective function was employed, and the minimization was carried out with m = 1 and over the prediction horizon, p. The polynomial structure of the system model yielded a polynomial objective function for the NMPC problem. Differentiation of this scalar objective function with respect to the manipulated input variables provided the first-order necessary conditions for optimality: a set of nu (equal to the number of inputs) coupled polynomials. Via Gröbner basis transformation, this set of polynomials was converted to a structured set of higher-order polynomials solvable via roots calculations and back-substitution. The algorithm was tested using a two-input two-output polymerization case study.

Solving nonconvex optimization problems

It is well known that most analysis and design problems in robust and nonlinear control can be formulated as global optimization problems with polynomial objective functions and constraints. The objective of this article is to show how GloptiPoly can solve challenging nonconvex optimization problems in robust nonlinear control. GloptiPoly is a general-purpose software with a user-friendly interface. First, we describe several nonconvex optimization problems arising in control system analysis and design. These problems involve multivariable polynomial objective functions and constraints. We then review the theoretical background behind GloptiPoly. Next, an example is presented to illustrate the successive linear matrix inequality relaxations and general features of the GloptiPoly software. Finally, we apply GloptiPoly to several control-related problems.

Optimum Design of Reinforced Concrete Beams Using Polynomial Optimization Technique

Structural optimization seeks the selection of design variables to achieve within the limit (constraints) placed on the structural behaviour, geometry or other factors; its goal of optimality defined by the objective function for specified loading conditions. The three basic features design variables, objective function and constraints contrive to form the design problem. There are several mathematical techniques to solve such problems. The polynomial optimization technique is a recently evolved procedure which is concerned with finding the minimum of a polynomial objective function subjected to constraints. A structural design problem has been formulated in this manner which enables minimum cost design to be derived rapidly and simply. A paper deals with the application of Polynomial optimization technique to R.C.C. beam-member design problem. In the present study this technique is used to determine the minimum cost of reinforced concrete members by considering several design variables such as breadth, depth, area of reinforcing steel etc. Since it is difficult for the designer in the office to become familiar with the mathematical computation required, further attempt is made to represent the resulting optimum design expressions in the form of “Nomograms” which will facilitate the work in the design office.

Global Optimization of Nonconvex Polynomial Programming Problems HavingRational Exponents

This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.

Polynomial objective functions for flash calculations: binary, ternary, and quaternary systems

Polynomial objective functions are developed for the vapor fraction in vapor-liquid equilibrium flash calculations for binary, ternary, and quaternary systems. The form of the polynomial suggests possible expansion to N-component systems. An extensive testing of these objective functions was performed using experimental data for binary systems. Comparative analysis with traditional Rachford-Rice objective function demonstrates better predictive capability and improved computational strategy for the new form. Comparison between predictions and experimental data shows a good match, even for cases where the Rachford-Rice objective function fails to yield realistic results.

Calibration of an objective function for the optimization of real‐time reservoir operations

Optimization models have been used as simulation tools to mimic complex decision making processes such as real‐time reservoir system operations. A recurring difficulty in the development of optimization models for use as simulation tools is the proper specification of the objective function. A new method for calibrating the objective function of an optimization model is presented. The method is developed specifically for the quadratic program class of models, but is applicable to general nonlinear programs with polynomial objective functions. A simplified case study is used to exercise the method in calibrating a quadratic objective function of a math program used to simulate operation of the Green River Basin hydrosystem in Kentucky. Results of the case study indicate that the method may be successfully used in certain cases to determine objective function parameter values to approximate decision making processes.

INVENTORY MODELS: OPTIMIZATION BY GEOMETRIC PROGRAMMING

ABSTRACTGeometric programming is a mathematical programming technique that is designed to determine the constrained minimum value of a generalized polynomial objective function. To date, most applications of the technique have been restricted to certain classes of engineering problems.This paper presents a brief summary of geometric programming and then illustrates its application to managerial problems by applying it to three well‐known inventory models.