A General Solution for the Errors in Variables (EIV) Model with Equality and Inequality Constraints

An improved mixed total least squares method for strain inversion from distance changes

A security-constrained power dispatch problem with non-convex cost function for a lossy electric power system area including limited energy supply thermal units fueled under a take-or-pay agreement is formulated. Then, an iterative solution method based on a modified subgradient algorithm based on feasible values and common pseudo scaling factor for limited energy supply thermal units is proposed and used to solve the problem. In the iterative proposed solution method, a modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the common pseudo scaling factor for limited energy supply thermal units is employed to adjust the amount of fuel consumed by the limited energy supply units during the considered operation period. Since all equality and inequality constraints in the non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles (the off-nominal tap settings), they are taken as independent variables except for those belonging to the reference bus. Load flow equations are added to the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, and off-nominal tap setting constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method given in this article before application of the modified subgradient algorithm based on feasible values to the optimization model. The proposed technique is tested on an IEEE 30-bus test system with a non-convex total cost rate curve. A portion of the dispatch problem considered in this article, where only a loading of the system in a subinterval is considered, was also solved by some other recently reported dispatch techniques. First, minimum total fuel cost rates and solution times obtained from the modified subgradient algorithm based on feasible values versus other techniques are compared, and the outperformance of the modified subgradient algorithm based on feasible values in terms of both total cost rate and solution time is demonstrated. After that, the optimal total fuel cost value and solution time of the system for the operation period for the cases where the fuel constraint is ignored and not ignored are presented, and it is shown that adding the fuel constraint into the dispatch problem further decreases the optimal total fuel cost for the considered dispatch problem. Finally, the modified subgradient algorithm based on feasible values was applied to the fuel constrained dispatch problem directly. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem.

In this paper, energy efficient power control for small cells underlaying a macro cellular network is investigated. We formulate the power control problem in self-organizing small cell networks as a non-cooperative game, and propose a distributed energy efficient power control scheme, which allows the small base stations (SBSs) to take individual decisions for attaining the Nash equilibrium (NE) with minimum information exchange. Specially, in the non-cooperative power control game, a non-convex optimization problem is formulated for each SBS to maximize their energy efficiency (EE). By exploiting the properties of parameter-free fractional programming and the concept of perspective function, the non-convex optimization problem for each SBS is transformed into an equivalent constrained convex optimization problem. Then, the constrained convex optimization problem is converted into an unconstrained convex optimization problem by exploiting the mixed penalty function method. The inequality constraints are eliminated by introducing the logarithmic barrier functions and the equality constraint is eliminated by introducing the quadratic penalty function. We also theoretically show the existence and the uniqueness of the NE in the non-cooperative power control game. Simulation results show remarkable improvements in terms of EE by using the proposed scheme.

Adaptive modeling strategy for constrained global optimization with application to aerodynamic wing design

Solving combinatorial optimization problems of the kind that can be codified by quadratic unconstrained binary optimization (QUBO) is a promising application of quantum computation. Some problems of this class suitable for practical applications such as the traveling salesman problem (TSP), the bin packing problem (BPP), or the knapsack problem (KP) have inequality constraints that require a particular cost function encoding. The common approach is the use of slack variables to represent the inequality constraints in the cost function. However, the use of slack variables considerably increases the number of qubits and operations required to solve these problems using quantum devices. In this work, we present an alternative method that does not require extra slack variables and consists of using an unbalanced penalization function to represent the inequality constraints in the QUBO. This function is characterized by larger penalization when the inequality constraint is not achieved than when it is. We evaluate our approach on the TSP, BPP, and KP, successfully encoding the optimal solution of the original optimization problem near the ground state cost Hamiltonian. Additionally, we employ D-Wave Advantage and D-Wave hybrid solvers to solve the BPP, surpassing the performance of the slack variables approach by achieving solutions for up to 29 items, whereas the slack variables approach only handles up to 11 items. This new approach can be used to solve combinatorial problems with inequality constraints with a reduced number of resources compared to the slack variables approach using quantum annealing or variational quantum algorithms.

A chebyshev polynomial method for optimal control with state constraints

This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints testH:h(β) ≥ 0 versusK: β ∈RK. We extend these results to consider local hypotheses tests of the formH:h1(β) ≥ 0 andh2(β) = 0 versusK: β ∈RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of χ2-distributions. We contrast local as opposed to global inequality constraints testing and give conditions on the model and constraints when each is possible. This paper also extends the well-known duality results in testing multivariate equality constraints to the case of nonlinear multivariate inequality constraints and combinations of nonlinear inequality and equality constraints.

A variable reduction strategy for evolutionary algorithms handling equality constraints

Traditional methods for handling (inequality) constraints in the quantum approximate optimization (QAOA) typically rely on penalty terms and slack variables, which increase problem complexity and expand the search space. More sophisticated mixer-based QAOA variants restrict the search within the feasible assignments but often suffer from prohibitive circuit complexity. This paper presents a penalty-free formalism for incorporating inequality constraints into the cost function of QAOA using an oracle-based subroutine that evaluates constraint satisfaction in an additional register, subsequently called indicator function QAOA (IF-QAOA). Applied to the knapsack problem, we demonstrate in numerical simulations the superior performance of IF-QAOA over conventional penalty-based approaches. Using advanced QAOA simulation techniques, we find that IF-QAOA achieves significantly higher solution quality and a faster time to solution in 82% of our benchmark cases, even though circuit depth is approximately three times larger. Analysis of the scaling behavior shows favorable scaling of IF-QAOA compared to penalty-based methods. Also, benchmarks against the recently developed quantum tree generator QAOA for knapsack problems (P. Christiansen , ) demonstrated higher solution quality for circuits of similar depth. Additionally, this paper introduces a method for approximating the indicator function when the number of ancillary qubits is limited or the constraint function is noninteger. With a specialized simulation algorithm based on projective measurements, we empirically demonstrate that this formalism can encode general inequality constraints using a fixed number of ancillary qubits.

The present paper is devoted to the application of the space transformation technique for solving linear and nonlinear programming problems. By using a surjective mapping the initial constrained optimization problem is transformed to a problem in a new space with only equality constraints. For the numerical solution of the latter problem we use the generalized gradient-projection method and Newton’s method. After inverse transformation to the initial space we obtain the family of numerical methods for solving optimization problems with equality and inequality constraints. In the linear programming case after some simplification and after choosing a particular exponential transformation function we obtain from the proposed methods the well-known primal-dual interior point linear programming algorithm and the so-called “variation of Kermarkar’s algorithm”.

In the present paper, we explore issues in handling equality constraints in multiobjective robust design optimization (RDO) problems. Satisfying an equality constraint exactly under uncertainty can be a challenging task. The challenge of handling equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is a complex task by itself. Equality constraints add to this complexity because of the additional tradeoffs that are introduced between constraint satisfaction under uncertainty and multiobjective performance. Satisfying equality constraints at their mean values, an approach typically followed, could lead to undesirable results. In this paper, we study the implications of equality constraint satisfaction in a multiobjective RDO problem, and provide a new problem formulation approach to resolve the above discussed tradeoffs. We illustrate that the proposed formulation can be used as an effective multiobjective design space exploration tool, with emphasis on equality constraint satisfaction under uncertainty. We present numerical examples to illustrate our theoretical developments.

We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set. The objective function is defined by a sum of local objective functions, while the global constraint set is produced by the intersection of local constraint sets. In particular, we study two cases: one where the equality constraint is absent, and the other where the local constraint sets are identical. We devise two distributed primal-dual subgradient algorithms based on the characterization of the primal-dual optimal solutions as the saddle points of the Lagrangian and penalty functions. These algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property, and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the Slater's condition.

In this paper, the classical exact absolute value penalty function method is used for solving a new class of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with both inequality and equality constraints are considered here, in which not all functions constituting them have the fundamental property of convex functions and most classes of generalized convex functions—namely, a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a Lagrange multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l1 penalty function. Further, under nondifferentiable ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l1 penalty function, is presented.

Some major challenges for geophysicists and structural geologists using three-dimensional boundary element method codes (3D-BEM) are: (1) reducing the amount of memory required to solve large and dense systems and (2) incorporation of inequality constraints such as traction inequality constraints (TIC) and displacement inequality constraints (DIC). The latter serves two purposes. First, for example, inequality constraints can be used to simulate frictional slip (using TIC). Second, these constraints can prevent element interpenetration while allowing opening mode (using DIC). We have developed a method that simultaneously incorporates both types of functionality of the inequality constraints. We show that the use of an appropriate iterative solver not only avoids the allocation of significant memory for solving the system (allowing very large model computation and simplifying parallelization on multi-core processors), but also admits interesting features such as natural incorporation of TICs and DICs. Compared to other techniques of contact management (e.g., Lagrange multipliers, penalty method, or complementarity problem), this new simple methodology, which does not use any incremental trial-and-error procedures, brings more flexibility, while making the system more stable and less subject to round-off errors without any computational overhead. We provide validations and comparisons of the inequality constraints implementation using 2D analytical and numerical solutions.

In this paper, we focus on the inequality constrained optimization problems, where the involved functions belong to the class , i.e. functions which have Lipschitz continuous gradient mappings. We propose a second-order mean value inequality for a function in terms of its limiting second-order subdifferential. By virtue of this inequality, using the second-order tangent set and the asymptotic second-order tangent cone, we establish the second-order Fritz John type necessary and sufficient optimality conditions for differentiable optimization problems with inequality and set constraints.