Functional Inequality Constraints Research Articles

A recurrent neural network approach for nonconvex interval quadratic programming

In the paper, a new approach is presented to tackle an interval optimization problem with nonconvex interval objective function and affine interval equality and inequality constraints utilizing a recurrent neural network. The approach involves converting the nonconvex interval quadratic programming into a cubic optimization via convex combinations and considering its weight as a decision variable. The recurrent neural network is then established to solve the cubic optimization through subgradient techniques and an exterior penalty function method. Moreover, it is proven that there is a unique global solution to the recurrent neural network, and its trajectory can reach the feasible region in a finite time and remain there. Additionally, it is demonstrated that the trajectory of recurrent neural network converges exponentially or in finite time towards a singleton belonging to the set of critical points of the cubic optimization. This approach is beneficial for interval quadratic programming problems where the weight is unknown or challenging to determine in practical applications. Finally, the effectiveness of this approach is verified through two numerical examples.

Distributed strategies for mixed equilibrium problems: Continuous-time theoretical approaches

In this paper, the mixed equilibrium problem is studied by employing a continuous-time multi-agent system, where the objective of agents is to cooperatively find a point from the feasible set such that the sum of bifunctions with a free-variable is nonnegative. Different from existing works on mixed equilibrium problems with only convex set constraints, we consider a more general case where the feasible set is constrained by a set of convex inequalities. Moreover, each agent only has access to the information of its own bifunction and inequality constraint function, as well as the information of a local convex set. To handle this problem, the Karush–Kuhn–Tucker condition that is suitable for continuous-time theoretical approaches is provided, and a continuous-time distributed primal–dual algorithm is proposed. Under mild assumptions on the graph and bifunctions, we prove that the states of all agents asymptotically converge to the common solution of the mixed equilibrium problem. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.

Gravity inversion method using L0-norm constraint with auto-adaptive regularization and combined stopping criteria

Abstract. We present a gravity inversion method that can produce compact and sharp images to assist the modeling of non-smooth geologic features. The proposed iterative inversion approach makes use of L0-norm-stabilizing functional, hard and physical parameter inequality constraints and a depth-weighting function. The method incorporates an auto-adaptive regularization technique, which automatically determines a suitable regularization parameter and error-weighting function that helps to improve both the stability and convergence of the method. The auto-adaptive regularization and error-weighting matrix are not dependent on the known noise level. Because of that, the method yields reasonable results even if the noise level of the data is not known properly. The utilization of an effectively combined stopping rule to terminate the inversion process is another improvement that is introduced in this work. The capacity and the efficiency of the new inversion method were tested by inverting randomly chosen synthetic and measured data. The synthetic test models consist of multiple causative blocky bodies, with different geometries and density distributions that are vertically and horizontally distributed adjacent to each other. Inversion results of the synthetic data show that the developed method can recover models that adequately match the real geometry, location and densities of the synthetic causative bodies. Furthermore, the testing of the improved approach using published real gravity data confirmed the potential and practicality of the method in producing compact and sharp inverse images of the subsurface.

Optimal control of continuous stochastic systems with functional inequality constraints

We investigate the problem of optimal control of nonlinear stochastic systems whose mathematical model is described by an ordinary Ito stochastic differential equation with functional inequality constraints on the trajectory. Analogues of the stochastic Pontryagin maximum principle, the linearized maximum condition are established.

On regularization of the Lagrange principle in the optimization problems for linear distributed Volterra type systems with operator constraints

Regularization of the classical optimality conditions - the Lagrange principle and the Pontryagin maximum principle - in a convex optimal control problem subject to functional equality and inequality constraints is considered. The controlled system is described by a linear functional-operator equation of second kind of the general form in the space $L_2^m$. The main operator on the right-hand side of the equation is assumed to be quasi-nilpotent. The objective functional to be minimized is strongly convex. The derivation of the regularized classical optimality conditions is based on the use of the dual regularization method. The main purpose of the regularized Lagrange principle and regularized Pontryagin maximum principle is to stably generate minimizing approximate solutions in the sense of J. Warga. As an application of the results obtained for the general linear functional-operator equation of second kind, two examples of concrete optimal control problems related to a system of delay equations and to an integro-differential transport equation are discussed.

О регуляризации недифференциальной теоремы Куна-Таккера в нелинейной задаче на условный экстремум

We consider a regular parametric nonlinear (nonconvex) problem for constrained extremum with an operator equality constraint and a finite number of functional inequality constraints. The constraints of the problem contain additive parameters, which makes it possible to use the apparatus of the “nonlinear” perturbation method for its study. The set of admissible elements of the problem is a complete metric space, and the problem itself may not have a solution. The regularity of the problem is understood in the sense that it has a generalized Kuhn-Tucker vector. Within the framework of the ideology of the Lagrange multiplier method, a regularized nondifferential Kuhn-Tucker theorem is formulated and proved, the main purpose of which is the stable generation of generalized minimizing sequences in the problem under consideration. These minimizing sequences are constructed from subminimals (minimals) of the modified Lagrange function taken at the values of the dual variable generated by the corresponding regularization procedure for the dual problem. The construction of the modified Lagrange function is a direct consequence of the subdifferential properties of a lower semicontinuous and, generally speaking, nonconvex value function as a function of the problem parameters. The regularized Kuhn-Tucker theorem “overcomes” the instability properties of its classical counterpart, is a regularizing algorithm, and serves as a theoretical basis for creating algorithms of practical solving problems for constrained extremum.

Distributed Resource Allocation Algorithm for General Linear Multiagent Systems

We focus on the optimal resource allocation problems with global equality constraints and convex function inequality constraints over heterogeneous linear multi-agent systems. The resource allocation problem aims to minimize the total objective function through neighboring information exchange. First, based on the state variable information, Karush-Kuhn-Tucker(KKT) conditions, and proportional-integral control concept, we propose an initialization-free distributed optimization algorithm, where each agent is driven by the gradient(subgradient) of its local objective function and local constraint convex function. In addition, the penalty factor control parameter is changed adaptively. Next, we propose an output-based distributed optimization algorithm that uses a Luenberger observer when the state variable is not accessible. Based on Lyapunov stability, it is proved that the proposed algorithms converge to the optimal solution of the resource allocation problem. Finally, simulation examples are used to demonstrate the effectiveness of the proposed algorithms.

Optimal control of a stochastic diffusion process with functional equality and inequality constraints

We consider a problem of optimal control of a multidimensional process specified by Ito stochastic differential equation, in the presence of functional equality and inequality constraints on the phase trajectory, with the control acting on the drift coefficient. Sufficiently general and at the same time constructively verifiable first- and second-order necessary optimality conditions are formulated in terms of variations of functionals.

Новый подход к математическому моделированию календарного планирования на промышленном предприятии

The purpose of this work was to build a model of industrial plant management and develop a schedule plan based on it. The paper considers a controlled model of product flows at an industrial enterprise. The model describes the interaction between various objects at the enterprise: production plants, manifolds, overpasses, piers, transport. In addition, the model specifies the composition and proportions of the components of raw materials and additives for each type of manufactured product, as well as physical constraints on the objects involvedin the production process. This brings us to the model of both the quality functional and equality and inequality constraints in the mathematical formulation. After building the model, the original control problem was reduced to two problems: direct and inverse. The numerical solution of both is provided in a close relationship between them. The study uses the findings from the theory of mathematical modeling, linear algebra and optimization theory, and leads to the development of the technique of building models of scheduling at an enterprise and numerical methods for solving the corresponding management problems. The research results are of great practical importance and can be used for effective management of the production process in many industries

Generalized Optimality Criteria Method for Topology Optimization

In this article, a generalized optimality criteria method is proposed for topology optimization with arbitrary objective function and multiple inequality constraints. This algorithm uses sensitivity information to update both the Lagrange multipliers and design variables. Different from the conventional optimality criteria method, the proposed method does not satisfy constraints at every iteration. Rather, it improves the Lagrange multipliers and design variables such that the optimality criteria are satisfied upon convergence. The main advantages of the proposed method are its capability of handling multiple constraints and computational efficiency. In numerical examples, the proposed method was found to be more than 100 times faster than the optimality criteria method and more than 1000 times faster than the method of moving asymptotes.

Strong Convergence on the Aggregate Constraint-Shifting Homotopy Method for Solving General Nonconvex Programming

In the paper, the aggregate constraint-shifting homotopy method for solving general nonconvex nonlinear programming is considered. The aggregation is only about inequality constraint functions. Without any cone condition for the constraint functions, the existence and convergence of the globally convergent solution to the K-K-T system are obtained for both feasible and infeasible starting points under much weaker conditions.

Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming

ABSTRACTSecond-order optimality conditions for the vector nonlinear programming problems with inequality constraints and continuously differentiable data are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce notions of a second-order pseudoconvex vector function and KKT-pseudoconvex vector problem with inequality constraints. We derive necessary and sufficient conditions for efficiency in the vector problem with inequality constraints. Three examples are presented.

Minimization of PAPR for DFT-Spread OFDM With BPSK Symbols

In this paper, the problem of reducing the peak-to-average power ratio (PAPR) of a discrete Fourier transform spread orthogonal frequency-division multiplexing signal is considered for the transmission of BPSK symbols. In particular, a joint optimization of the constellation-rotation angle and the pulse-shaping vector is conducted with or without the allocation of additional sub-carriers. To avoid the exponential complexity required to evaluate the PAPR, a computationally efficient upper bound on the worst-case PAPR is derived and adopted as the objective function. Unlike conventional upper bounds, the proposed upper bound exploits the structure in the relative phase among symbol waveforms and incorporates the constellation-rotation effect. The signal-to-interference-plus-noise ratio (SINR) at the output of a constellation-derotating matched filter is also derived along with a generalized Nyquist criterion and adopted as the inequality constraint function. Then, joint optimizations are conducted through numerical search for various signal parameters and target SINRs. Surprisingly, the optimal rotation angle is neither 0 nor $\pi /2$ . It is shown that the proposed combinations significantly outperform the existing combinations of the rotation angle and the shaping vector when the signal is amplified by a nonlinear power amplifier.

Primal–Dual Mirror Descent Method for Constraint Stochastic Optimization Problems

Extension of the mirror descent method developed for convex stochastic optimization problems to constrained convex stochastic optimization problems (subject to functional inequality constraints) is studied. A method that performs an ordinary mirror descent step if the constraints are insignificantly violated and performs a mirror descent step with respect to the violated constraint if this constraint is significantly violated is proposed. If the method parameters are chosen appropriately, a bound on the convergence rate (that is optimal for the given class of problems) is obtained and sharp bounds on the probability of large deviations are proved. For the deterministic case, the primal–dual property of the proposed method is proved. In other words, it is proved that, given the sequence of points (vectors) generated by the method, the solution of the dual method can be reconstructed up to the same accuracy with which the primal problem is solved. The efficiency of the method as applied for problems subject to a huge number of constraints is discussed. Note that the bound on the duality gap obtained in this paper does not include the unknown size of the solution to the dual problem.

On Nonconvex Optimization Problems with D.C. Equality and Inequality Constraints

Abstract The paper addresses the nonconvex nonsmooth optimization problem with the cost function and equality and inequality constraints given by d.c. functions. The original problem is reduced to a problem without constraints with the help of the exact penalization theory. After that, the penalized problem is represented as a d.c. minimization problem without constraints, for which the new mathematical tools under the form of global optimality conditions (GOCs) are developed. The GOCs reduce the nonconvex problem in question to a family of convex (linearized with respect to the basic nonconvexities) problems. In addition, the GOCs are related to some nonsmooth form of the KKT-theorem for the original problem. On the base of the developed theory we propose new numerical methods for local and global search.

Mathematical model for determination of the most advantageous conditions for formation of parts of aerospace engineering on the operations of the end milling

A mathematical model has been introduced for determining the most advantageous conditions for parts formation when end milling operations. This model consists of a linear objective function and linear inequality constraints and takes into account the kinetics of thermal processes in the cutting zone. The equation determines the processing time was used as the objective function and constraint inequalities are related with the functional parameters and the parameters of the milling process and determines the quality of the machining.

Mathematical model for determining rational machining conditions for flat grinding with the wheel periphery on machines with a rectangular table

A mathematical model is presented for determining the rational machining conditions for flat grinding operations by the rim of a wheel on machines with a rectangular table consisting of a linear objective function and linear inequality constraints. As the objective function, the equation, determining the main machining time, was used. And constraints which are related to the functional parameters and parameters determining the machining quality and the kinematic capabilities of the machine were used as inequality constraints.

Dual form reduction in the atomic optimization method

One-dimensional optimization problems with a polynomial objective function and polynomial matrix inequality constraints are considered. For problems dual to their linear relaxations, a transformation is presented that makes them compatible with the atomic optimization method, both in its basic and in the generalized form with a reduced number of atoms.

Constraint qualifications in nonsmooth multiobjective optimization problem

A multiobjective optimization problem (MOP) with inequality and equality constraints is considered where the objective and inequality constraint functions are locally Lipschitz and equality constraint functions are differentiable. Burachik and Rizvi [J. Optim. Theory Appl. 155, 477-491 (2012)] gave Guignard and generalized Abadie regularity conditions for a differentiable programming problem and derived Karush-Kuhn-Tucker (KKT) type necessary optimality conditions. In this paper, we have defined the nonsmooth versions of Guignard and generalized Abadie regularity conditions given by Burachik and Rizvi and obtained KKT necessary optimality conditions for efficient and weak efficient solutions of (MOP). Further several constraint qualifications sufficient for the above newly defined constraint qualifications are introduced for (MOP) with no equality constraints. Relationships between them are presented and examples are constructed to support the results.

Stable iterative Lagrange principle in convex programming as a tool for solving unstable problems

A convex programming problem in a Hilbert space with an operator equality constraint and a finite number of functional inequality constraints is considered. All constraints involve parameters. The close relation of the instability of this problem and, hence, the instability of the classical Lagrange principle for it to its regularity properties and the subdifferentiability of the value function in the problem is discussed. An iterative nondifferential Lagrange principle with a stopping rule is proved for the indicated 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. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimization problems is discussed. The capabilities of this principle are illustrated by numerically solving the classical ill-posed problem of finding the normal solution of a Fredholm integral equation of the first kind.