Presence Of Inequality Constraints Research Articles

Robust Launch Vehicle Trajectory Optimization with Stage Impact and Heat Flux Constraints

A fast trajectory optimization/guidance algorithm in the presence of path constraints for a multistage launch vehicle is presented in this work. The optimization problem is to maximize the payload to be injected into a geostationary transfer orbit with constraints on argument of perigee, second-stage impact point longitude, and heat flux constraint on the third stage that flies a low-altitude trajectory. These constraints constitute a unique set of equality and inequality constraints as they are contradictory in nature. Most of the gradient-based trajectory optimization schemes require an initial guess close enough to the optimal value for assured convergence. In addition, the rate of convergence deteriorates in the presence of inequality constraints. These issues are addressed by providing a solution to the optimization problem with low sensitivity to the initial guess profiles. A self-starting optimization scheme that can generate initial guess profiles automatically from boundary conditions is the major contribution of this work. The proposed method presented in this paper also gives a methodology for real-time implementation of the optimization scheme in the onboard computer. The sensitivity to initial guess in a gradient-based optimization procedure, in the presence of inequality constraints, is clearly brought out as part of this work by studying the convergence from random sets of initial guess profiles. The real-time implementation of the algorithm is validated with various initial conditions in deterministic as well as Monte Carlo sense to establish the robustness of the algorithm.

Stability of a system of circular rings reinforced with inextensible threads

The paper deals with the problems of stability for a system of circular rings interconnected in such a way that the displacements of these rings, and the rotation angles of their sections at some points, coincide. This has been reduced to a certain variational problem with restrictions on the functions in question in the form of linear equations, and Fourier series are used to obtain a finite-dimensional approximation. The paper also presents the stability problem for a system of circular rings reinforced by inextensible threads that cannot resist compressive forces. In this case, constraints in the form of inequalities arise, and after finite-dimensional approximation the problem reduces to finding the bifurcation points for a nonlinear programming problem in the presence of inequality constraints.

Задачи устойчивости круговых арок с неудерживающими связями

The work deals with the problems of stability of arches and arch systems supported by inextensible threads that cannot withstand compressive forces. These questions relate to contact problems of elastic structures with an unknown region of active interaction of their elements, and the study is reduced to the search and analysis of bifurcation points ofsolutions to variational problems in the presence of inequality constraints on the sought functions. The paper considers the problem of stability of an arch supported by threads, one end of which is attached to some point of the arch, the other to the projection of this point on the chord connecting its ends. In the numerical solution, the problem is reducedto finding a load parameter in which the nonconvex quadratic programming problem with constraints on variables in the form of inequalities has a non-unique solution. Numerical calculations have shown that even with a small number of reinforcing threads, the critical load approximately doubles.

Projection-Free Distributed Optimization With Nonconvex Local Objective Functions and Resource Allocation Constraint

We present a novel generalized constrained convex optimization model for multiagent systems that contains both the local, coupled equality, and inequality constraints, and a global resource allocation constraint. This model unifies the traditional constrained optimization problem, the resource allocation problem, and the economic dispatch problem. Unlike the majority of literature where each local objective function is required to be convex, we only require a milder condition that the global objective function is convex. The gradient of the global Lagrangian is estimated locally by each agent using the dynamic average consensus protocol. Synchronously, modified primal-dual dynamics produce the optimal solution via the estimated gradient. The generalized Lagrange multiplier method is introduced to avoid the usual positive projections in the presence of inequality constraints. This leads to smooth dynamics and a continuous Lyapunov derivative, which enables the exponential stability analysis. Simulation examples support the proposed distributed methods.

ADMM‐based algorithm for solving DC‐OPF in a large electricity network considering transmission losses

The authors address the problem of solving DC-optimal power flow (OPF) considering transmission losses in a large electricity network. The loss in a line is considered in the power balance equation and is taken as proportional to the absolute value of the flow through the line. Many standard solvers fail to converge to an optimal solution of the DC-OPF for comparatively large bus systems, even with a quadratic cost of generation. The authors use a decomposition algorithm such as alternating directions method of multipliers (ADMM) to address this problem. However, the ADMM algorithm cannot be directly applied to this problem because of the sparsity of the coefficient matrices of the objective function and the presence of inequality constraints. Thus, the authors introduce two relaxations to the DC-OPF problem, namely the regularisation and the modified penalisation. The authors provide a novel ADMM algorithm for the regularised and the modified penalised problem which converges to an optimal solution even for large bus systems. The authors show that the ADMM algorithm converges near to the optimal solution of the DC-OPF problem if the regularisation and modified penalisation parameters are chosen carefully. Numerical simulations illustrate the effectiveness of the algorithm.

Sobol' indices for problems defined in non-rectangular domains

A novel theoretical and numerical framework for the estimation of Sobol’ sensitivity indices for models in which inputs are confined to a non-rectangular domain (e.g., in presence of inequality constraints) is developed. Two numerical methods, namely the quadrature integration method which may be very efficient for problems of low dimensionality and the MC/QMC estimators based on the acceptance-rejection sampling method are proposed for the numerical estimation of Sobol’ sensitivity indices. Several model test functions with constraints are considered for which analytical solutions for Sobol’ sensitivity indices were found. These solutions were used as benchmarks for verifying numerical estimates. The method is shown to be general and efficient.

An Auxiliary Particle Filtering Algorithm With Inequality Constraints

For nonlinear non-Gaussian stochastic dynamic systems with inequality state constraints, this technical note presents an efficient particle filtering algorithm, constrained auxiliary particle filtering algorithm. To deal with the state constraints, the proposed algorithm probabilistically selects particles such that those particles far away from the feasible area are less likely to propagate into the next time step. To improve on the sampling efficiency in the presence of inequality constraints, it uses a highly effective method to perform a series of constrained optimization so that the importance distributions are constructed efficiently based on the state constraints. The caused approximation errors are corrected using the importance sampling method. This ensures that the obtained particles constitute a representative sample of the true posterior distribution. A simulation study on vehicle tracking is used to illustrate the proposed approach.

Solving constrained quadratic binary problems via quantum adiabatic evolution

Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.

Fluid pressure arrival-time tomography: Estimation and assessment in the presence of inequality constraints with an application to production at the Krechba field, Algeria

Deformation in the overburden proves useful in deducing spatial and temporal changes in the volume of a producing reservoir. Based on these changes, we have estimated diffusive traveltimes associated with the transient flow due to production, and then, as the solution of a linear inverse problem, the effective permeability of the reservoir. An advantage of the approach based on traveltimes, as opposed to one based on the amplitude of surface deformation, is that it is much less sensitive to the exact geomechanical properties of the reservoir and overburden. Inequalities constrain the inversion, under the assumption that the fluid production only results in pore volume decreases within the reservoir. The formulation has been applied to satellite-based estimates of deformation in the material overlying a thin gas production zone at the Krechba field in Algeria. The peak displacement after three years of gas production is found to be approximately [Formula: see text], overlying the eastern margin of the anticlinal structure defining the gas field. Using data from 15 irregularly spaced images of range change, we have calculated the diffusive traveltimes associated with the startup of a gas production well. The inequality constraints were incorporated into the estimates of model parameter resolution and covariance, improving the resolution by roughly 30% to 40%.

Global Optimality Conditions for Discrete and Nonconvex Optimization—With Applications to Lagrangian Heuristics and Column Generation

The well-known and established global optimality conditions based on the Lagrangian formulation of an optimization problem are consistent if and only if the duality gap is zero. We develop a set of global optimality conditions that are structurally similar but are consistent for any size of the duality gap. This system characterizes a primal–dual optimal solution by means of primal and dual feasibility, primal Lagrangian ε-optimality, and, in the presence of inequality constraints, a relaxed complementarity condition analogously called δ-complementarity. The total size ε + δ of those two perturbations equals the size of the duality gap at an optimal solution. Further, the characterization is equivalent to a near-saddle point condition which generalizes the classic saddle point characterization of a primal–dual optimal solution in convex programming. The system developed can be used to explain, to a large degree, when and why Lagrangian heuristics for discrete optimization are successful in reaching near-optimal solutions. Further, experiments on a set-covering problem illustrate how the new optimality conditions can be utilized as a foundation for the construction of new Lagrangian heuristics. Finally, we outline possible uses of the optimality conditions in column generation algorithms and in the construction of core problems.

Iksim: A fast algorithm for indicator kriging and simulation in the presence of inequality constraints, hard and soft data

iksim is a fast program for indicator kriging and simulation of continuous or categorical variables in the presence of inequality constraints, hard and soft data. The speed advantage of iksim program results from the FFT spectral simulation of the non-conditional p-field. To maintain the full speed of median IK in the presence of missing indicator values corresponding to constraint intervals or soft data, the missing values can be reconstructed using the marginal z-distribution. The algorithm, program structure, data file formats, parameter file, and some validation checks are presented.

Quadratic programming applications in finance using Excel

The paper describes two applications of quadratic programming in finance, one from the early years (Markowitz's efficient portfolios with minimum risk) and the other a more recent innovation (Sharpe's style analysis which estimates an implied asset allocation for an investment fund). We show how, in the presence of inequality constraints, Excel's Solver can be used to find the optimal weights in both quadratic programming applications. We also implement a direct analytic solution for generating the efficient frontier when there are no inequality constraints using the matrix functions in Excel. Both applications use only a small number of asset classes and require repeated use of the minimisation task. We show how Visual Basic for Applications (Microsoft's macro language for Excel) can be used to program such tasks, confirming that techniques that were the preserve of dedicated software only a few years ago can now be easily replicated using Excel to solve real problems.

Constrained observer-based controller for linear continuous-time systems

Abstract The observer-based controller design in the presence of inequality constraints on the system input is considered. The solution is obtained by an adequate use of the positive invariance concept

Identity Between INS Position and Velocity Error Models

Conclusions While optimal controls extremalize only a given scalar performance index, researchers have been understandably reluctant to abandon the formalism in the presence of inequality constraints which induce an additional quadratic optimality criterion with a constant weight. The performance index may then inhibit a preferred throttle operation over a time interval. In applications allowing obvious physical interpretations, for example, maximum allowable control effort could be prolonged to reduce a terminal state deviation; to do so, however, would disallow using the convenient Riccati equation formalism in most linear optimization applications published to date. Replacement of constant weights by time-varying weighting functions, surprisingly rare in applications literature, is advocated here to permit usage of the full linear formalism while keeping the performance index selections more consistent with the true intent of the optimization. The rudimentary example and approach to synthesizing a time varying weighting function presented here are intended to stimulate further development for broader application. Acknowledgment The authors are indebted to Miss Cherle Williams for digital program coding.

On optimality conditions in nondifferentiable programming

This paper is devoted to necessary optimality conditions in a mathematical programming problem without differentiability or convexity assumptions on the data. The main tool of this study is the concept of generalized gradient of a locally Lipschitz function (and more generally of a lower semi-continuous function). In the first part, we consider local extremization problems in the unconstrained case for objective functions taking values in (źź, +ź]. In the second part, the constrained case is considered by the way of the cone of adherent displacements. In the presence of inequality constraints, we derive in the third part optimality conditions in the Kuhn--Tucker form under a constraint qualification.