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This work introduces a new Riemannian optimization method for registering open parameterized surfaces with a constrained global optimization approach. The proposed formulation leads to a rigorous theoretic foundation and guarantees the existence and the uniqueness of a global solution. We also propose a new Bayesian clustering approach where local distributions of surfaces are modeled with spherical Gaussian processes. The maximization of the posterior density is performed with Hamiltonian dynamics which provide a natural and computationally efficient spherical Hamiltonian Monte Carlo sampling. Experimental results demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, rigid body static optimization is investigated on the Riemannian manifold of rigid body motion groups. This manifold, which is also a matrix manifold, provides a framework to formulate translational and rotational motions of the body, while considering any coupling between those motions, and uses members of the special orthogonal group SO (3) to represent the rotation. Hence, it is called the special Euclidean group SE (3) . Formalism of rigid body motion on SE (3) does not fall victim to singularity or non-uniqueness issues associated with attitude parameterization sets. Benefiting from Riemannian matrix manifolds and their metrics, a generic framework for unconstrained static optimization and a customizable framework for constrained static optimization are proposed that build a foundation for dynamic optimization of rigid body motions on SE (3) and its tangent bundle. The study of Riemannian manifolds from the perspective of rigid body motion introduced here provides an accurate tool for optimization of rigid body motions, avoiding any biases that could otherwise occur in rotational motion representation if attitude parameterization sets were used. [ABSTRACT FROM AUTHOR]

In this paper, an improved retractable body model (IRBM) is established, which has an advantage in simulating the flexion-and-extension motion of skier's legs during carved turning and straight gliding. The trajectory optimization problem for the nonlinear alpine skiing system is transformed into a multi-phase optimal control (MPOC) problem. Subsequently, a constrained multi-phase trajectory optimization model is developed based on the optimal control theory, where the optimization target is to minimize the total skiing time. The optimization model is discretized by using the Radau pseudospectral method (RPM), which transcribes the MPOC problem into a nonlinear programming (NLP) problem that is then solved by SNOPT solver. Through numerical simulations, the optimization results under different constraints are obtained using MATLAB. The variation characteristics of the variables and trajectories are analyzed, and four influencing factors related to the skiing time are investigated by comparative experiments. It turns out that the small turning radius can reduce the total skiing time, the flexion-and-extension motion of legs is beneficial to skier's performance, and the large inclination angle can shorten skier's turning time, while the control force has a slight effect on the skiing time. The effectiveness and feasibility of the proposed models and trajectory optimization strategies are validated by simulation and experiment results. [ABSTRACT FROM AUTHOR]

We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher's augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches ε -approximate second-order critical points of the original optimization problem in at most O (ε - 3) iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher's augmented Lagrangian, which may be of independent interest. [ABSTRACT FROM AUTHOR]

This paper proposes a novel Difference-of-Convex (DC) decomposition for polynomials using a power-sum representation, achieved by solving a sparse linear system. We introduce the Boosted DCA with Exact Line Search ( BDCA e ) for addressing linearly constrained polynomial programs within the DC framework. Notably, we demonstrate that the exact line search equates to determining the roots of a univariate polynomial in an interval, with coefficients being computed explicitly based on the power-sum DC decompositions. The subsequential convergence of BDCA e to critical points is proven, and its convergence rate under the Kurdyka–Łojasiewicz property is established. To efficiently tackle the convex subproblems, we integrate the Fast Dual Proximal Gradient method by exploiting the separable block structure of the power-sum DC decompositions. We validate our approach through numerical experiments on the Mean–Variance–Skewness–Kurtosis portfolio optimization model and box-constrained polynomial optimization problems. Comparative analysis of BDCA e against DCA, BDCA with Armijo line search, UDCA, and UBDCA with projective DC decomposition, alongside standard nonlinear optimization solvers FMINCON and FILTERSD, substantiates the efficiency of our proposed approach. [ABSTRACT FROM AUTHOR]

We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method, is for solving Riemannian constrained optimization problems. We establish its local superlinear and quadratic convergence under the standard assumptions. Moreover, we show its global convergence when it is combined with a classical line search. Our method is a generalization of the classical framework of primal-dual interior point methods for nonlinear nonconvex programming. Numerical experiments show the stability and efficiency of our method. [ABSTRACT FROM AUTHOR]

Orthogonally decomposable (odeco) tensors is a special class of symmetric tensors. Previous works have focused on investigating its E-eigenpairs problem, and made some theoretical achievements concerning the number and the local optimality of E-eigenpairs. However, concerning local optimality of each eigenpair, the existing work only analyzed the third-order tensor case. In this paper, we further exploit this issue for any higher-order tensors by checking second-order necessary condition of the related constrained optimization model and deducing an equivalent matrix formula criterion for local optimality identification. Finally, a generalized conclusion for local optimality of eigenpairs for odeco tensors is provided, and some simulated experiments are conducted for validation. [ABSTRACT FROM AUTHOR]

Tokamaks are the most promising devices to prove the feasibility of energy production using nuclear fusion on Earth which is foreseen as a possible source of energy for the next centuries. In large tokamaks with superconducting poloidal field (PF) coils, the problem of avoiding saturation of the currents is of paramount importance, especially for a reactor such as the European demonstration fusion power plant DEMO. Indeed, reaching the current limits during plasma operation may cause a loss of control of the plasma shape and/or current, leading to a major disruption. Therefore, a current limit avoidance (CLA) system is essential to assure safe operation. Three different algorithms to be implemented within a CLA system are proposed in this paper: two are based on online solutions of constrained optimization problems, while the third one relies on dynamic allocation. The performance assessment for all the proposed solutions is carried out by considering challenging operation scenarios for the DEMO reactor, such as the case where more than one PF current simultaneously saturates during the discharge. An evaluation of the computational burden needed to solve the allocation problem for the various proposed alternatives is also presented, which shows the compliance of the optimization-based approaches with the envisaged deadlines for real-time implementation of the DEMO plasma magnetic control system. [ABSTRACT FROM AUTHOR]

The paper provides strong convergence of solutions to a sequence of linear-quadratic (LQ) optimization problems defined in an abstract functional framework. Each problem is accompanied by the constraint of reaching a given target within a prescribed precision. We show that the problems are well-posed and characterize their solutions. The main result provides the conditions under which these solutions converge to the minimizer of the limit problem. The generality of the result allows its application to a wide range of problems: elliptic, parabolic, control ones, etc. The examples presented in the paper consider optimal approximate controls of the heat equation and optimal approximate solutions to the Poisson equation. [ABSTRACT FROM AUTHOR]

The main objective of this works is to present an efficient hybrid optimization approach using a new coupling technique for solving constrained engineering design problems. This hybrid is based on the simulated annealing algorithm with the projected gradient and its stochastic perturbation. The proposed hybrid is combined with corrected techniques in order to correct the solutions out of domain and send them to the domain's border. The proposed algorithm is tested and evaluated on several benchmark functions, as well as on the basis of some engineering design problems. The obtained results are well compared with typical approaches existing in the literature. The solutions obtained by the proposed hybrid are more accurate than those given by other known methods and the performance and efficiency of the proposed algorithm are demonstrated. [ABSTRACT FROM AUTHOR]

Focusing on smooth constrained optimization problems, and inspired by the complementary approximate Karush–Kuhn–Tucker (CAKKT) conditions, this work introduces the weighted complementary approximate Karush–Kuhn–Tucker (WCAKKT) conditions. They are shown to be verified by limit points generated not only by safeguarded augmented Lagrangian methods, but also by inexact restoration methods, inverse and logarithmic barrier methods, and a penalized algorithm for constrained nonsmooth optimization. Under the analyticity of the feasible set description, and resting upon a desingularization result, the new conditions are proved to be equivalent to the CAKKT conditions. The WCAKKT conditions capture the algebraic elements of the desingularization result needed to characterize CAKKT sequences using a weighted complementarity condition that asymptotically sums zero. Due to its generality and strength, the new condition may help to enlighten the practical performance of algorithms in generating CAKKT sequences. [ABSTRACT FROM AUTHOR]

This paper investigates a private distributed optimization problem over a multi-agent network, where the goal is to cooperatively minimize the sum of all locally convex cost functions subject to coupled equality constraints over time-varying unbalanced directed networks while considering privacy concerns. To solve this problem, we integrate push-sum protocols with dual subgradient methods to design a private distributed dual stochastic push-sum algorithm. Under the assumption of convexity, we first establish the convergence of the algorithm in terms of dual variables, primal variables and constraint violations. Then we show that the algorithm has a sub-linear growth with order of O (ln t / t) . The result reveals that there is a tradeoff between the privacy level and the accuracy of the algorithm. Finally, the efficiency of the algorithm is verified numerically over two applications to the economic dispatch problems and electric vehicles charging control problems. [ABSTRACT FROM AUTHOR]

The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas–Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas–Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem. [ABSTRACT FROM AUTHOR]

In this paper, we study a general minimization vector problem which is expressed in terms of a perturbation mapping defined on a product of locally convex Hausdorff topological vector spaces with values in another locally convex topological vector space. Several representations of the epigraph of the conjugate of the perturbation mapping are given, and then, variants vector Farkas lemmas associated with the system defined by this mapping are established. A dual problem and another so-called loose dual problem of the mentioned problem are defined and stable strong duality results between these pairs of primal–dual problems are established. The results just obtained are then applied to a general class of composed constrained vector optimization problems. For this class of problems, two concrete perturbation mappings are proposed. These perturbation mappings give rise to variants of dual problems including the Lagrange dual problem and several kinds of Fenchel–Lagrange dual problems of the problem under consideration. Stable strong duality results for these pairs of primal–dual problems are derived. Several classes of concrete vector (and scalar) optimization problems are also considered at the end of the paper to illustrate the significance of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the minimization of an indefinite quadratic function over the intersection of balls and linear inequality constraints (QOBL). Using the hyperplanes induced by the intersection of each pair of balls, we show that the optimal solution of QOBL can be found by solving several extended trust-region subproblems (e-TRS). To solve e-TRS, we use the alternating direction method of multipliers approach and a branch and bound algorithm. Numerical experiments show the efficiency of the proposed approach compared to the CVX and the extended adaptive ellipsoid-based algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonlinear optimization problems with nonlinear equality constraints and bound constraints on the variables. For the solution of such problems, many augmented Lagrangian methods have been defined in the literature. Here, we propose to modify one of these algorithms, namely ALGENCAN by Andreani et al., in such a way to incorporate second-order information into the augmented Lagrangian framework, using an active-set strategy. We show that the overall algorithm has the same convergence properties as ALGENCAN and an asymptotic quadratic convergence rate under suitable assumptions. The numerical results confirm that the proposed algorithm is a viable alternative to ALGENCAN with greater robustness. [ABSTRACT FROM AUTHOR]

This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach consists of reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class by using generalized differentiation. In this way, we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of first-order necessary optimality conditions for such and related problems in broad spaces of functions including those of class C k as k ≥ 1 . [ABSTRACT FROM AUTHOR]

We present the second-order point-based (necessary and sufficient) optimality conditions for nonlinear programming with continuously differentiable data, via the regular and limiting (Mordukhovich) second-order subdifferentials. The sharper results are obtained for C 1 , 1 data. Also, we derive a second-order characterization for (strictly and strongly) pseudoconvex, continuously differentiable functions. [ABSTRACT FROM AUTHOR]

In the paper, we consider both finite-dimensional and infinite-dimensional optimization problems with inclusion-type and equality-type constraints. We obtain sufficient conditions for the stability in the weak topology of a solution to this problem with respect to small perturbations of the problem parameters. In the finite-dimensional case, conditions for the stability in the strong topology of the solution are obtained for the problem with equality-type constraints. These conditions are based on a certain implicit function theorem. [ABSTRACT FROM AUTHOR]

This paper proposes a multi-scale method to design a continuous-time distributed algorithm for constrained convex optimization problems by using multi-agents with Markov switched network dynamics and noisy inter-agent communications. Unlike most previous work which mainly puts emphasis on dealing with fixed network topology, this paper tackles the challenging problem of investigating the joint effects of stochastic networks and the inter-agent communication noises on the distributed optimization dynamics, which has not been systemically studied in the past literature. Also, in sharp contrast to previous work in constrained optimization, we depart from the use of projected gradient flow which is non-smooth and hard to analyze; instead, we design a smooth optimization dynamics which leads to easier convergence analysis and more efficient numerical simulations. Moreover, the multi-scale method presented in this paper generalizes previously known distributed convex optimization algorithms from the fixed network topology to the switching case and the stochastic averaging obtained in this paper is a generalization of the existing deterministic averaging. [ABSTRACT FROM AUTHOR]

We study a class of deterministic two-player nonzero-sum differential games where one player uses piecewise-continuous controls to affect the continuously evolving state, while the other player uses impulse controls at certain discrete instants of time to shift the state from one level to another. The state measurements are made at some given instants of time, and players determine their strategies using the last measured state value. We provide necessary and sufficient conditions for the existence of sampled-data Nash equilibrium for a general class of differential games with impulse controls. We specialize our results to a scalar linear-quadratic differential game and show that the equilibrium impulse timing can be obtained by determining a fixed point of a Riccati-like system of differential equations with jumps coupled with a system of nonlinear equality constraints. By reformulating the problem as a constrained nonlinear optimization problem, we compute the equilibrium timing, and level of impulses. We find that the equilibrium piecewise continuous control and impulse control are linear functions of the last measured state value. Using a numerical example, we illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a projected subgradient method for solving constrained nondifferentiable quasiconvex multiobjective optimization problems. The algorithm is based on the Plastria subdifferential to overcome potential shortcomings known from algorithms based on the classical gradient. Under suitable, yet rather general assumptions, we establish the convergence of the full sequence generated by the algorithm to a Pareto efficient solution of the problem. Numerical results are presented to illustrate our findings. [ABSTRACT FROM AUTHOR]

A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided. [ABSTRACT FROM AUTHOR]

We study the implications of a well-known metric inequality condition on sets to the applicability of standard necessary optimality conditions for constrained optimization problems when a new constraint is added. We compare this condition with several other constraint qualification conditions in the literature, and due to its metric nature, we apply it to nonsmooth optimization problems in order to perform first a penalization and then to give optimality conditions in terms of generalized differentiability. [ABSTRACT FROM AUTHOR]

We propose a local convergence analysis of a primal–dual interior point algorithm for the solution of a bound-constrained optimization problem. The algorithm includes a regularization technique to prevent singularity of the matrix of the linear system at each iteration, when the second-order sufficient conditions do not hold at the solution. These conditions are replaced by a milder assumption related to a local error-bound condition. This new condition is a generalization of the one used in unconstrained optimization. We show that by an appropriate updating strategy of the barrier parameter and of the regularization parameter, the proposed algorithm owns a superlinear rate of convergence. The analysis is made thanks to a boundedness property of the inverse of the Jacobian matrix arising in interior point algorithms. An illustrative example is given to show the behavior and the gain obtained by this regularization strategy. [ABSTRACT FROM AUTHOR]

In this paper, optimality conditions are studied for a class of constrained interval-valued optimization problems governed by path-independent curvilinear integral (mechanical work) cost functionals. Specifically, a minimal criterion of optimality for a local LU-optimal solution of the considered PDE&PDI-constrained variational control problem to be its global LU-optimal solution is formulated and proved. In addition, the main result is highlighted by an illustrative application describing the controlled behavior of an artificial neural system. [ABSTRACT FROM AUTHOR]

In this work, a multi-scale optimization strategy for lightweight structures, based on a global-local modelling approach is presented. The approach is applied to a realistic wing structure of a civil aircraft. The preliminary design of the wing can be formulated as a constrained optimization problem, involving several requirements at the different scales of the structure. The proposed strategy is characterized by two main features. Firstly, the problem is formulated in the most general sense, by including all design variables involved at each problem scale. Secondly, two scales are considered: (i) the structure macroscopic scale, where low-fidelity numerical models are used; (ii) the structure mesoscopic scale (or component-level), where enhanced models are involved. In particular, the structural responses are evaluated at both global and local scales, avoiding the use of approximated analytical methods. To this end, fully parametric global and local finite element models are interfaced with an in-house genetic algorithm. Refined models are created only for the most critical regions of the structure, and linked to the global one by means of a dedicated sub-modelling approach. [ABSTRACT FROM AUTHOR]

This paper pursues a twofold goal. Firstly, we aim at deriving novel second-order characterizations of important robust stability properties of perturbed Karush–Kuhn–Tucker systems for a broad class of constrained optimization problems generated by parabolically regular sets. Secondly, the obtained characterizations are applied to establish well-posedness and superlinear convergence of the basic sequential quadratic programming method to solve parabolically regular constrained optimization problems. [ABSTRACT FROM AUTHOR]

Motivated by a growing list of nontraditional statistical estimation problems of the piecewise kind, this paper provides a survey of known results supplemented with new results for the class of piecewise linear-quadratic programs. These are linearly constrained optimization problems with piecewise linear-quadratic objective functions. We first summarize some local properties of a piecewise linear-quadratic function in terms of their first- and second-order directional derivatives. We next extend some well-known necessary and sufficient second-order conditions for local optimality of a quadratic program to a piecewise linear-quadratic program and provide a dozen such equivalent conditions for strong, strict, and isolated local optimality, showing in particular that a piecewise linear-quadratic program has the same characterizations for local minimality as a standard quadratic program. As a consequence of one such condition, we show that the number of strong, strict, or isolated local minima of a piecewise linear-quadratic program is finite; this result supplements a recent result about the finite number of directional stationary objective values. We also consider a special class of unconstrained composite programs involving a non-differentiable norm function, for which we show that the task of verifying the second-order stationary condition can be converted to the problem of checking the copositivity of certain Schur complement on the nonnegative orthant. [ABSTRACT FROM AUTHOR]

This paper introduces a method for computing points satisfying the second-order necessary optimality conditions for nonconvex minimization problems subject to a closed and convex constraint set. The method comprises two independent steps corresponding to the first- and second-order conditions. The first-order step is a generic closed map algorithm, which can be chosen from a variety of first-order algorithms, making it adjustable to the given problem. The second-order step can be viewed as a second-order feasible direction step for nonconvex minimization subject to a convex set. We prove that any limit point of the resulting scheme satisfies the second-order necessary optimality condition, and establish the scheme's convergence rate and complexity, under standard and mild assumptions. Numerical tests illustrate the proposed scheme. [ABSTRACT FROM AUTHOR]

We combine the Lagrangian dual decomposition, barrier smoothing, path-following, and proximal Newton techniques to develop a new inexact interior-point Lagrangian decomposition method to solve a broad class of constrained composite convex optimization problems. Our method allows one to approximately solve the primal subproblems (called the slave problems), which leads to inexact oracles (i.e., inexact function value, gradient, and Hessian) of the smoothed dual problem (called the master problem). By appropriately controlling the inexact computation in both the slave and master problems, we can still establish a polynomial-time iteration complexity of our algorithm and recover primal solutions. We illustrate the performance of our method through two numerical examples and compare it with existing methods. [ABSTRACT FROM AUTHOR]

In this study, a novel sequential optimality condition for general continuous optimization problems is established. In the context of mathematical programs with equilibrium constraints, the condition is proved to ensure Clarke stationarity. Originally devised for constrained nonsmooth optimization, the proposed sequential optimality condition addresses the domain of the constraints instead of their images, capturing indistinctly the features of the complementarity and the ordinary constraints of optimization problems modeling equilibrium conditions. [ABSTRACT FROM AUTHOR]

Inspired by recent work on safe feature elimination for 1-norm regularized least-squares, we develop strategies to eliminate features from convex optimization problems with non-negativity constraints. Our strategy is safe in the sense that it will only remove features/coordinates from the problem when they are guaranteed to be zero at a solution. To perform feature elimination, we use an accurate, but not optimal, primal–dual feasible pair, making our methods robust and able to be used on ill-conditioned problems. We supplement our feature elimination problem with a method to construct an accurate dual feasible point from an accurate primal feasible point; this allows us to use a first-order method to find an accurate primal feasible point and then use that point to construct an accurate dual feasible point and perform feature elimination. Under reasonable conditions, our feature elimination strategy will eventually eliminate all zero features from the problem. As an application of our methods, we show how safe feature elimination can be used to robustly certify the uniqueness of nonnegative least-squares problems. We give numerical examples on a well-conditioned synthetic nonnegative least-squares problem and on a set of 40,000 extremely ill-conditioned problems arising in a microscopy application. [ABSTRACT FROM AUTHOR]

We present a novel fully adaptive conditional gradient method with the step length regulation for solving pseudo-convex constrained optimization problems. We propose some deterministic rules of the step length regulation in a normalized direction. These rules guarantee to find the step length by utilizing the finite procedures and provide the strict relaxation of the objective function at each iteration. We prove that the sequence of the function values for the iterates generated by the algorithm converges globally to the objective function optimal value with sublinear rate. [ABSTRACT FROM AUTHOR]

In this paper, we aim at applying improvement sets and image space analysis to investigate scalarizations and optimality conditions of the constrained set-valued optimization problem. Firstly, we use the improvement set to introduce a new class of generalized convex set-valued maps. Secondly, under suitable assumptions, some scalarization results of the constrained set-valued optimization problem are obtained in the sense of (weak) optimal solution characterized by the improvement set. Finally, by considering two classes of nonlinear separation functions, we present the separation between two suitable sets in image space and derive some optimality conditions for the constrained set-valued optimization problem. It shows that the existence of a nonlinear separation is equivalent to a saddle point condition of the generalized Lagrangian set-valued functions. [ABSTRACT FROM AUTHOR]

This paper proposes a partially inexact proximal alternating direction method of multipliers for computing approximate solutions of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic iteration-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact alternating direction method of multipliers with relative error criteria have been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method. [ABSTRACT FROM AUTHOR]

In this paper, an analytical expression of the optimal solution for a box-constrained linear symmetric cone optimization problem is proposed. The resulting theories are established based on the theory of the spectral decomposition of a symmetric cone. Moreover, we apply our results to develop algorithms for solving several symmetric cone optimization problems and conduct some preliminary numerical experiments to show the performance of the developed algorithms. [ABSTRACT FROM AUTHOR]

We present a primal-dual augmented Lagrangian method for solving an equality constrained minimization problem, which is able to rapidly detect infeasibility. The method is based on a modification of the algorithm proposed in Armand and Omheni (Optim Methods Softw 32(1):1-21, 2017). A new parameter is introduced to scale the objective function and, in case of infeasibility, to force the convergence of the iterates to an infeasible stationary point. It is shown, under mild assumptions, that whenever the algorithm converges to an infeasible stationary point, the rate of convergence is quadratic. This is a new convergence result for the class of augmented Lagrangian methods. The global convergence of the algorithm is also analyzed. It is also proved that, when the algorithm converges to a stationary point, the properties of the original algorithm are preserved. The numerical experiments show that our new approach is as good as the original one when the algorithm converges to a local minimum, but much more efficient in case of infeasibility. [ABSTRACT FROM AUTHOR]

For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well defined and necessarily converges to this specific solution (despite degeneracy, and despite that there are other solutions nearby). We note that unlike the common settings of convergence analyses, our assumptions subsume that a local Lipschitzian error bound does not hold for the solution in question. Our results apply to constrained and projected variants of the Gauss-Newton, Levenberg-Marquardt, and LP-Newton methods. Applications to smooth and piecewise smooth reformulations of complementarity problems are also discussed. [ABSTRACT FROM AUTHOR]

A generalized fractional programming problem is defined as the problem of minimizing a nonlinear function, defined as the maximum of several ratios of functions on a feasible domain. In this paper, we propose new methods based on the method of centers, on the proximal point algorithm and on the idea of bundle methods, for solving such problems. First, we introduce proximal point algorithms, in which, at each iteration, an approximate prox-regularized parametric subproblem is solved inexactly to obtain an approximate solution to the original problem. Based on this approach and on the idea of bundle methods, we propose implementable proximal bundle algorithms, in which the objective function of the last mentioned prox-regularized parametric subproblem is replaced by an easier one, typically a piecewise linear function. The methods deal with nondifferentiable nonlinearly constrained convex minimax fractional problems. We prove the convergence, give the rate of convergence of the proposed procedures and present numerical tests to illustrate their behavior. [ABSTRACT FROM AUTHOR]

In this paper, a general methodology to approximate sets of data points through Non-uniform Rational Basis Spline (NURBS) curves is provided. The proposed approach aims at integrating and optimizing the full set of design variables (both integer and continuous) defining the shape of the NURBS curve. To this purpose, a new formulation of the curve fitting problem is required: it is stated in the form of a constrained nonlinear programming problem by introducing a suitable constraint on the curvature of the curve. In addition, the resulting optimization problem is defined over a domain having variable dimension, wherein both the number and the value of the design variables are optimized. To deal with this class of constrained nonlinear programming problems, a global optimization hybrid tool has been employed. The optimization procedure is split in two steps: firstly, an improved genetic algorithm optimizes both the value and the number of design variables by means of a two-level Darwinian strategy allowing the simultaneous evolution of individuals and species; secondly, the optimum solution provided by the genetic algorithm constitutes the initial guess for the subsequent gradient-based optimization, which aims at improving the accuracy of the fitting curve. The effectiveness of the proposed methodology is proven through some mathematical benchmarks as well as a real-world engineering problem. [ABSTRACT FROM AUTHOR]

In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to address new approaches, in separate ways, to necessary and, respectively, sufficient optimality conditions in constrained vector optimization. In this respect, for the necessary optimality conditions that we derive, we use a kind of vectorial penalization technique, while for the sufficient optimality conditions we make use of an appropriate scalarization method. In both cases, the approaches couple a basic technique (of penalization or scalarization, respectively) with several results in variational analysis and optimization obtained by the authors in the last years. These combinations allow us to arrive to optimality conditions which are, in terms of assumptions made, new. [ABSTRACT FROM AUTHOR]

In this contribution, a new methodology based on a double-loop iteration process is proposed for the treatment of uncertainties in engineering system design. The inner optimization loop is used to find the solution associated with the highest probability value (inverse reliability analysis), and the outer loop is the regular optimization loop used to solve the considered reliability problem through differential evolution and multi-objective optimization differential evolution algorithms. The proposed methodology is applied to mathematical functions and to the design of classical engineering systems according to both mono- and multi-objective contexts. The obtained results are compared with those obtained by classical approaches and demonstrate that the proposed strategy represents an interesting alternative to reliability design of engineering systems. [ABSTRACT FROM AUTHOR]

In this paper, an optimal feedback, for a free vibrating semi-active controlled plant, is derived. The problem is represented as a constrained optimal control problem of a single input, free vibrating bilinear system, and a quadratic performance index. It is solved by using Krotov's method and to this end, a novel sequence of Krotov functions that suits the addressed problem, is derived. The solution is arranged as an algorithm, which requires solving the states equation and a differential Lyapunov equation in each iteration. An outline of the proof for the algorithm convergence is provided. Emphasis is given on semi-active control design for stable free vibrating plants with a single control input. It is shown that a control force, derived by the proposed technique, obeys the physical constraint related with semi-active actuator force without the need of any arbitrary signal clipping. The control efficiency is demonstrated with a numerical example. [ABSTRACT FROM AUTHOR]

This paper proposes a self-adaptive penalty function and presents a penalty-based algorithm for solving nonsmooth and nonconvex constrained optimization problems. We prove that the general constrained optimization problem is equivalent to a bound constrained problem in the sense that they have the same global solutions. The global minimizer of the penalty function subject to a set of bound constraints may be obtained by a population-based meta-heuristic. Further, a hybrid self-adaptive penalty firefly algorithm, with a local intensification search, is designed, and its convergence analysis is established. The numerical experiments and a comparison with other penalty-based approaches show the effectiveness of the new self-adaptive penalty algorithm in solving constrained global optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush-Kuhn-Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush-Kuhn-Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush-Kuhn-Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality. [ABSTRACT FROM AUTHOR]

We present a primal-dual algorithm for solving a constrained optimization problem. This method is based on a Newtonian method applied to a sequence of perturbed KKT systems. These systems follow from a reformulation of the initial problem under the form of a sequence of penalized problems, by introducing an augmented Lagrangian for handling the equality constraints and a log-barrier penalty for the inequalities. We detail the updating rules for monitoring the different parameters (Lagrange multiplier estimate, quadratic penalty and log-barrier parameter), in order to get strong global convergence properties. We show that one advantage of this approach is that it introduces a natural regularization of the linear system to solve at each iteration, for the solution of a problem with a rank deficient Jacobian of constraints. The numerical experiments show the good practical performances of the proposed method especially for degenerate problems. [ABSTRACT FROM AUTHOR]

We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied. [ABSTRACT FROM AUTHOR]

Stochastic search methods for global optimization and multi-objective optimization are widely used in practice, especially on problems with black-box objective and constraint functions. Although there are many theoretical results on the convergence of stochastic search methods, relatively few deal with black-box constraints and multiple black-box objectives and previous convergence analyses require feasible iterates. Moreover, some of the convergence conditions are difficult to verify for practical stochastic algorithms, and some of the theoretical results only apply to specific algorithms. First, this article presents some technical conditions that guarantee the convergence of a general class of adaptive stochastic algorithms for constrained black-box global optimization that do not require iterates to be always feasible and applies them to practical algorithms, including an evolutionary algorithm. The conditions are only required for a subsequence of the iterations and provide a recipe for making any algorithm converge to the global minimum in a probabilistic sense. Second, it uses the results for constrained optimization to derive convergence results for stochastic search methods for constrained multi-objective optimization. [ABSTRACT FROM AUTHOR]