Functional Inequality Constraints Research Articles

A Simple Parallel Algorithm with an $O(1/t)$ Convergence Rate for General Convex Programs

This paper considers convex programs with a general (possibly nondifferentiable) convex objective function and Lipschitz continuous convex inequality constraint functions. A simple algorithm is developed and achieves an $O(1/t)$ convergence rate. Similar to the classical dual subgradient algorithm and the alternating direction method of multipliers (ADMM) algorithm, the new algorithm has a parallel implementation when the objective and constraint functions are separable. However, the new algorithm has a faster $O(1/t)$ convergence rate compared with the best known $O(1/\sqrt{t})$ convergence rate for the dual subgradient algorithm with primal averaging. Further, it can solve convex programs with nonlinear constraints, which cannot be handled by the ADMM algorithm. The new algorithm is applied to a multipath network utility maximization problem and yields a decentralized flow control algorithm with the fast $O(1/t)$ convergence rate.

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.

A branch and bound algorithm for quantified quadratic programming

The aim of this paper is to find the global solutions of uncertain optimization problems having a quadratic objective function and quadratic inequality constraints. The bounded epistemic uncertainties in the constraint coefficients are represented using either universal or existential quantified parameters and interval parameter domains. This approach allows to model non-controlled uncertainties by using universally quantified parameters and controlled uncertainties by using existentially quantified ones. While existentially quantified parameters could be equivalently considered as additional variables, keeping them as parameters allows maintaining the quadratic problem structure, which is essential for the proposed algorithm. The branch and bound algorithm presented in the paper handles both universally and existentially quantified parameters in a homogeneous way, without branching on their domains, and uses some dedicated numerical constraint programming techniques for finding a robust, global solution. Several examples clarify the theoretical parts and the tests demonstrate the usefulness of the proposed method.

Optimization of three-loop missile autopilot gain under crossover frequency constraint

Abstract The open-loop crossover frequency is pointed as an important parameter for practical autopilot design. Since different gain designs may achieve the same open-loop crossover frequency, it should be neither considered as a performance objective of the optimal autopilot design-schemes nor neglected. Besides, the main assignment of the autopilot is to drive the missile to track the acceleration commands, so the autopilot gain design should be evaluated directly according to the resultant tracking performance. For this purpose, an optimal design methodology of the three-loop missile autopilot is introduced based on constraint optimization technique, where the tracking performance is established analytically as the design objective and the open-loop crossover frequency is formed as inequality constraint function, both are manipulated in terms of stable characteristic parameters of the autopilot closed-loop. The proposed technique is implemented with the assistance of a numerical optimization algorithm which automatically adjusts the design parameters. Finally, numerical simulation results are provided to demonstrate the effectiveness and feasibility of the proposed approach compared with that in some references.

The stable Lagrange principle in sequential form for the convex programming problem in a uniformly convex space and its applications

We consider the convex programming problem in a reflexive space with operator equality constraint and finite number of functional inequality constraints. For this problem we prove the stable with respect to the errors in the initial data Lagrange principle in sequential nondifferential form. We show that the sequential approach and dual regularization significantly expand a class of optimization problems that can be immediately and stably solved on a base of the classical design of the Lagrange function. We discuss the possibility of its applicability for solving unstable optimization problems.

A generalization of Löwner-John’s ellipsoid theorem

We address the following generalization $$\mathbf {P}$$ of the Löwner-John ellipsoid problem. Given a (non necessarily convex) compact set $$\mathbf {K}\subset \mathbb {R}^n$$ and an even integer $$d\in \mathbb {N}$$ , find an homogeneous polynomial $$g$$ of degree $$d$$ such that $$\mathbf {K}\subset \mathbf {G}:=\{\mathbf {x}:g(\mathbf {x})\le 1\}$$ and $$\mathbf {G}$$ has minimum volume among all such sets. We show that $$\mathbf {P}$$ is a convex optimization problem even if neither $$\mathbf {K}$$ nor $$\mathbf {G}$$ are convex! We next show that $$\mathbf {P}$$ has a unique optimal solution and a characterization with at most $${n+d-1\atopwithdelims ()d}$$ contacts points in $$\mathbf {K}\cap \mathbf {G}$$ is also provided. This is the analogue for $$d>2$$ of Löwner-John’s theorem in the quadratic case $$d=2$$ , but importantly, we neither require the set $$\mathbf {K}$$ nor the sublevel set $$\mathbf {G}$$ to be convex. More generally, there is also an homogeneous polynomial $$g$$ of even degree $$d$$ and a point $$\mathbf {a}\in \mathbb {R}^n$$ such that $$\mathbf {K}\subset \mathbf {G}_\mathbf {a}:=\{\mathbf {x}:g(\mathbf {x}-\mathbf {a})\le 1\}$$ and $$\mathbf {G}_\mathbf {a}$$ has minimum volume among all such sets (but uniqueness is not guaranteed). Finally, we also outline a numerical scheme to approximate as closely as desired the optimal value and an optimal solution. It consists of solving a hierarchy of convex optimization problems with strictly convex objective function and Linear Matrix Inequality constraints.

Optimal Design of Cosine Modulated Nonuniform Linear Phase FIR Filter Bank via Both Stretching and Shifting Frequency Response of Single Prototype Filter

This paper designs an optimal cosine modulated nonuniform linear phase finite impulse response (FIR) filter bank. The frequency responses of all the analysis filters and the synthesis filters of the filter bank are derived based on both stretching and shifting the frequency response of a single prototype filter. The total aliasing error of the filter bank is minimized subject to specifications on the maximum magnitude distortion of the filter bank and the maximum ripple magnitudes of the prototype filter over both the passband and the stopband. This paper proposes a joint constraint transcription and modified filled function method for solving the optimization problem. In particular, the functional inequality constraints are converted to discrete constraints via the constraint transcription method. The globally optimal solution of the nonconvex optimization problem can be found efficiently via the modified filled function method. Computer numerical simulation results show that our design outperforms existing designs.

Neural network for nonsmooth, nonconvex constrained minimization via smooth approximation.

A neural network based on smoothing approximation is presented for a class of nonsmooth, nonconvex constrained optimization problems, where the objective function is nonsmooth and nonconvex, the equality constraint functions are linear and the inequality constraint functions are nonsmooth, convex. This approach can find a Clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation. The global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded. Specially, the proposed network does not require: 1) the initial point to be feasible; 2) a prior penalty parameter to be chosen exactly; 3) a differential inclusion to be solved. Numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network.

On a Control Problem Containing Support Functions

A control problem containing support functions in the integrand of the objective of the functional as well as in the inequality constraint function is considered. For this problem, Fritz John and Karush-Kuhn-Tucker type necessary optimality conditions are derived. Using Karush-Kuhn-Tucker type optimality conditions, Wolfe type dual is formulated and usual duality theorems are established under generalized convexity conditions. Special cases are generated. It is also shown that our duality results have linkage with those of nonlinear programming problems involving support functions.

Stable sequential convex programming in a Hilbert space and its application for solving unstable problems

A parametric convex programming problem with an operator equality constraint and a finite set of functional inequality constraints is considered in a Hilbert space. The instability of this problem and, as a consequence, the instability of the classical Lagrange principle for it is closely related to its regularity and the subdifferentiability properties of the value function in the optimization problem. A sequential Lagrange principle in nondifferential form is proved for the indicated convex programming problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. It is shown that the classical Lagrange principle in this problem can be naturally treated as a limiting variant of its stable sequential counterpart. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimal control problems and inverse problems is discussed. For two illustrative problems of these kinds, the corresponding stable Lagrange principles are formulated in sequential form.

부등호 제약조건 처리 알고리즘 개발 및 전력계통 해석 적용

The optimal analysis has objective functions, equality constraint functions and inequality functions. Objective functions may be used with inequality function, because occasionally variables are moved to non-analytic condition with calculating objective functions. But inequality constraint functions are very complicated problem in a optimal analysis. this paper suggest a method to solve inequality constraint functions.

Sequential stable Kuhn-Tucker theorem in nonlinear programming

A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing sequence for this problem are formally constructed from elements of minimizing sequences for its augmented Lagrangian with values of dual variables chosen by applying the Tikhonov stabilization method in the course of solving the corresponding modified dual problem. A sequential Kuhn-Tucker theorem in nondifferential form is proved in terms of minimizing sequences and augmented Lagrangians. The theorem is stable with respect to errors in the initial data and provides a necessary and sufficient condition on the elements of a minimizing sequence. It is shown that the structure of the augmented Lagrangian is a direct consequence of the generalized differentiability properties of the value function in the problem. The proof is based on a “nonlinear” version of the dual regularization method, which is substantiated in this paper. An example is given illustrating that the formal construction of a minimizing sequence is unstable without regularizing the solution of the modified dual problem.

Simple Sequential Quadratically Constrained Quadratic Programming Feasible Algorithm with Active Identification Sets for Constrained Minimax Problems

In this paper, the nonlinear minimax problems with inequality constraints are discussed. Based on the idea of simple sequential quadratically constrained quadratic programming algorithm for smooth constrained optimization, an alternative algorithm for solving the discussed problems is proposed. Unlike the previous work, at each iteration, a feasible direction of descent called main search direction is obtained by solving only one subprogram which is composed of a convex quadratic objective function and simple quadratic inequality constraints without the second derivatives of the constrained functions. Then a high-order correction direction used to avoid the Maratos effect is computed by updating the main search direction with a system of linear equations. The proposed algorithm possesses global convergence under weak Mangasarian–Fromovitz constraint qualification and superlinear convergence under suitable conditions with the upper-level strict complementarity. At last, some preliminary numerical results are reported.

A Semidefinite Relaxation-Based Algorithm for Robust Attitude Estimation

This paper presents a tractable method for solving a robust attitude estimation problem, based on a weighted least squares approach with nonlinear constraints. Attitude estimation requires information of a few vector quantities, each obtained from both a sensor and a mathematical model. By considering the modeling errors, measurement noise, sensor biases and offsets as infinity-norm bounded uncertainties, we formulate a robust optimization problem, which is nonconvex with nonlinear cost and constraints. The robust min-max problem is approximated with a nonconvex minimization problem using an upper bound. A new regularization scheme is also proposed to improve the robust performance. We then use semidefinite relaxation to convert the suboptimal problem with quadratic cost and constraints into a tractable semidefinite program with a linear objective function and linear matrix inequality constraints. We also show how to extract the solution of the suboptimal robust estimation problem from the solution of the semidefinite relaxation. Further, a mathematical proof supported by numerical results is presented stating the gap between the suboptimal problem and its relaxation is zero under a given condition, which is mostly true in real life scenarios. The usefulness of the proposed algorithm in the presence of uncertainties is evaluated with the help of examples.

Reliability-based design optimization of an FPSO riser support using moving least squares response surface meta-models

The paper deals with the reliability-based design optimization (RBDO) of a riser support installed on a floating production storage and offloading (FPSO) unit under operation, extreme, damaged, and one line failure cases and installation loading conditions. The optimization problem is formulated such that probabilistic thickness variables described with random characteristics are determined by minimizing the weight of the riser support structure subjected to stress constraints for the given target reliability. The initial design model is generated based on actual FPSO riser support specifications. The finite element analysis is conducted using NASTRAN, and the probabilistic optimal solutions are obtained via the moving least squares method in the context of RBDO using a response surface meta-model. For the meta-modeling of the inequality constraint functions of stresses, a constraint-feasible moving least squares method (CF-MLSM) is adopted in the present study. The CF-MLSM has been shown to ensure constraint feasibility regardless of the nonlinearity of the constraint function, the feasible bounds, and the random characteristics during the meta-model-based RBDO process. The solution results from the proposed RBDO strategy present improved design performances under various riser operating conditions.

Reliability-based design optimization of knuckle component using conservative method of moving least squares meta-models

This paper discusses reliability-based design optimization (RBDO) of an automotive knuckle component under bump and brake loading conditions. The probabilistic design problem is to minimize the weight of a knuckle component subject to stresses, deformations, and frequency constraints in order to meet the given target reliability. The initial design is generated based on an actual vehicle specification. The finite element analysis is conducted using ABAQUS, and the probabilistic optimal solutions are obtained via the moving least squares method (MLSM) in the context of approximate optimization. For the meta-modeling of inequality constraint functions, a constraint-feasible moving least squares method (CF-MLSM) is used in the present study. The method of CF-MLSM based RBDO has been shown to not only ensure constraint feasibility in a case where the meta-model-based RBDO process is employed, but also to require low expense, as compared with both conventional MLSM and non-approximate RBDO methods.

One class of separable optimization problems: solution method, application

A solution method for solving optimization problems with a max-separable objective function and min-separable inequality constraints is suggested. The method is based on the results of 5, 6. Application in some location problems and small illustrative numerical examples are presented.

Optimum design of lattice-columns for buckling

This paper is about the problem of optimization of bracing columns--height of the parallel chords, cross-sectional areas of the chord and bracing and area moments of inertia of the chord and bracing. Optimization parameters have been determined by Lagrange's multipliers method--inequality constraint functions. Total mass of the construction has been selected as the objective function. Criterions of buckling (general buckling of the bracing column, local buckling of the chord and local buckling of the bracing diagonal) and criterion of deformation are used as the constraint functions. Numerical examples are performed by usage of the obtained theoretical solutions.

Tangent cone and contingent cone to the intersection of two closed sets

For a nonempty closed set C in a real normed vector space X and an inequality solution set, we present several sufficient conditions for the tangent and contingent cones to their intersection to contain the intersections of the corresponding cones. We not only express the contingent cone to a solution set of inequalities and equalities by the directional (or Fréchet) derivatives of the active inequality constraint functions and the Fréchet derivatives of the equality constraint functions but also the tangent cone by the Clarke (or lower Dini, or upper Dini) derivatives of the active inequality constraint functions and the directional derivatives of the equality constraint functions. By using a simple property of the function d C − d C c , we characterize these cones by the hypertangent and hypercontingent vectors to the set C . Furthermore, these results allow us to present new constraint qualifications for the Karush–Kuhn–Tucker conditions.

A response surface based sequential approximate optimization using constraint-shifting analogy

When a traditional response surface method (RSM) is used as a meta-model for inequality constraint functions, an approximate optimal solution is sometimes actually infeasible in a case where it is active at the constraint boundary. The paper proposes a new RSM that ensures the constraint feasibility with respect to an approximate optimal solution. Constraint-shifting is suggested in order to secure the constraint feasibility during the sequential approximate optimization process. A central composite design is used as a tool for design of experiments. The proposed approach is verified through a mathematical function problem and engineering optimization problems to support the proposed strategies.