Integral Inequality Constraints Research Articles

Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints

We introduce a new technique to optimize a linear cost function subject to a one-dimensional affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in stability analysis, input-to-state/output analysis, and control of many systems governed by partial differential equations (PDEs), in particular fluid dynamical systems. First, we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that under mild assumptions a convergent, non-decreasing sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can also be computed with SDPs. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our techniques by solving problems arising from the stability analysis of PDEs.

Efficient solution of the diesel-engine optimal control problem by time-domain decomposition

Offline solutions of the optimal control problems relating to diesel engines provide implications for causal control strategies and serve as a tool to automate the engine calibration process. However, the numerical solution of these dynamic optimisation problems is challenging. Above all, the model evaluations are expensive, integral and local inequality constraints are present, and the time horizon to be considered is long. Generally, the continuous problem is transcribed into a nonlinear mathematical program, which has a runtime that increases superlinearly with the problem size. In the literature, the time-domain decomposition of unconstrained and control-constrained problems is proposed to alleviate this often prohibiting increase in computational time. This paper extends the decomposition approach to problems with integral inequality constraints by utilising a partially dual formulation to derive equivalence conditions. To achieve these conditions, exact and approximate iteration schemes are proposed. The time-domain decomposition is combined with an iterative mesh refinement and a parallelisation of the model evaluations, which dominate the runtime of the solution process. On a standard personal computer, an overall speedup factor of 25 is achieved for a realistically sized instance of the diesel-engine problem.

Sufficient conditions for existence of physically significant solutions in limiting equilibrium slope stability analysis

Engineering assessment of slope stability is usually performed using limiting equilibrium analysis. This framework includes a process of minimization which identifies the critical slip surface and its associated minimal safety factor. The approach makes sense only if a minimum safety factor exists, i.e. if there is a slip surface for which the safety factor is smaller than safety factors associated with all other slip surfaces. The present work establishes conditions which guarantee that slope stability problems have a physically significant minimum. The question of existence of a minimum is relevant to all slope stability formulations which satisfy equilibrium conditions without a priori assumptions with respect to the shape of potential slip surfaces. The main purpose of the present work is to “legitimatize” the approximate, but practically useful, limiting equilibrium technique by placing it on secure foundations. The present work shows that the restrictions required in order to ensure the existence of a minimum include three, well motivated, physical elements: (a) Two integral inequality constraints restricting legitimate forms of slip surfaces, and normal stress functions acting on them. These constraints represent the obvious observation that under usual conditions slopes fail by moving down and away from the main body of the slope. (b) The strength model (Mohr envelope), should imply a finite tensile strength. (c) A “cracking criterion” which specifies the consequences (crack formation) occurring when the soil’s tensile strength is fully mobilized.

Optimization of a Class of Distributed Parameter Systems with Deviating Argument

In this paper we consider an optimal control problem described by a system of nonlinear first order hyperbolic partial differential equations with deviating argument, including integral inequality constraints. The control variables are assumed to be measurable, with the corresponding state variables in Lp. We introduce the adjoint equations, derive an integral representation of the increments of the functionals involved, and use separation theorems of functional analysis to obtain new necessary optimality conditions in the form of the Pontryagin maximum principle.

Necessary conditions of optimality in constrained planar domain optimization

The article considers optimization of the shape of a planar domain with an integral functional subject to integral inequality constraints and constraints on domain location. The boundary-value problem is a Dirichlet problem for a quasilinear elliptical equation. A method is proposed for deriving necessary conditions of optimality for problems of this kind.

Optimal control of systems governed by partial differential equations with integral inequality constraints

Optimal control of systems governed by partial differential equations with integral inequality constraints