The features that characterize the onset of Huntington disease (HD) are poorly understood yet have significant implications for research and clinical practice. Motivated by the need to address this issue, and the fact that there may be inaccuracies in clinical HD data, we apply robust optimization and duality techniques to study support vector machine (SVM) classifiers in the face of uncertainty in

We consider an optimal control problem for non-isothermal steady flows of low-concentrated aqueous polymer solutions in a bounded 3D domain. In this problem, the state functions are the flow velocity and the temperature, while the control function is the heat flux through a given part of the boundary of the flow domain. We obtain sufficient conditions for the existence of weak solutions that minimize

The relationships between Greenberg–Pierskalla’s subdifferential and some variants of it and semi-quasidifferentials of quasiconvex functions are studied in this paper. Also, some characterizations of quasiconvex functions in terms of semi-quasidifferentials and the connection between quasimonotonicity of semi-quasidifferentials and quasiconvexity are investigated.

A popular approach to recover low rank matrices is the nuclear norm regularized minimization (NRM) for which the selection of the regularization parameter is inevitable. In this paper, we build up a novel rule to choose the regularization parameter for NRM, with the help of the duality theory. Our result provides a safe set for the regularization parameter when the rank of the solution has an upper

I study a dynamic variant of the Dixit–Stiglitz (Am Econ Rev 67(3), 1977) model of monopolistic competition by introducing price stickiness à la Fershtman and Kamien (Econometrica 55(5), 1987). The analysis is restricted to bounded quantity and price paths that fulfill the necessary conditions for an open-loop Nash equilibrium. I show that there exists a symmetric steady state and that its stability

In this paper, we consider a new class of random differential variational inequalities (RDVIs) with nonlocal boundary conditions in Hilbert spaces. We apply the projection operator, Gronwall’s lemma and a result on the existence of a random differential inclusion to establish uniqueness and Hyers–Ulam stability results of the abstract inequality. As an illustrative application, the linear random differential

In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided conditions under which accumulation points satisfy a condition requiring a certain optimality function, used in the algorithms to generate search directions, to be nonnegative for all controls. The aim of this paper is to clarify the nature of this optimality

We introduce Stochastic Dynamic Cutting Plane (StoDCuP), an extension of the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve multistage stochastic convex optimization problems. At each iteration, the algorithm builds lower bounding affine functions not only for the cost-to-go functions, as SDDP does, but also for some or all nonlinear cost and constraint functions. We show the almost

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

The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees robust feasibility of an uncertain linear conic program. This determines when the robust feasible set is non-empty. Otherwise, the robust counterpart of an uncertain program is not well defined as a robust optimization problem. In this paper, we address a key fundamental question of

We study a general nonconvex formulation for low-rank minimization problems. We use recent results on non-Euclidean first-order methods to provide efficient and scalable algorithms. Our approach uses the geometry induced by the Bregman divergence of well-chosen kernel functions; for unconstrained problems, we introduce a novel family of Gram quartic kernels that improve numerical performance. Numerical

This paper presents a decentralized algorithm for non-convex optimization over tree-structured networks. We assume that each node of this network can solve small-scale optimization problems and communicate approximate value functions with its neighbors based on a novel multi-sweep communication protocol. In contrast to existing parallelizable optimization algorithms for non-convex optimization, the

The weighted complementarity problem (denoted by WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this paper, by introducing a one-parametric class of smoothing functions which includes the weight vector, we propose a smoothing Newton algorithm with nonmonotone line search to solve WCP. We show that

This paper studies the convergence of the Galerkin method and regularization for variational inequalities with pseudomonotone operators in the sense of Brézis. Namely, we prove that under certain conditions, the solutions of the Galerkin equations and regularized variational inequalities converge strongly to a solution of the original variational inequality in reflexive Banach spaces. An application

In this work, a novel strategy for the inverse optimal control of a class of affine nonlinear systems is proposed. The proposed strategy involves translation of the infinite horizon nonlinear optimal control problem into a Diagonal stability problem, followed by the formulation of a set of criteria for the synthesis of a stabilizing feedback control law. Hence, the proposed controller design methodology

This paper investigates online algorithms for smooth time-varying optimization problems, focusing first on methods with constant step-size, momentum, and extrapolation-length. Assuming strong convexity, precise results for the tracking iterate error (the limit supremum of the norm of the difference between the optimal solution and the iterates) for online gradient descent are derived. The paper then

There is a significant tendency in the industry for automation of the engineering design process. This requires the capability of analyzing an existing design and proposing or ideally generating an optimal design using numerical optimization. In this context, efficient and robust realization of such a framework for numerical shape optimization is of prime importance. Another requirement of such a framework

During the past few decades, multi-agent optimization problems have drawn increased attention from the research community. When multiple objective functions are present among agents, many works optimize the sum of these objective functions. However, this formulation implies a decision regarding the relative importance of each objective: optimizing the sum is a special case of a multi-objective problem

In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically

We consider a class of structured nonsmooth difference-of-convex minimization. We allow nonsmoothness in both the convex and concave components in the objective function, with a finite max structure in the concave part. Our focus is on algorithms that compute a (weak or standard) d(irectional)-stationary point as advocated in a recent work of Pang et al. (Math Oper Res 42:95–118, 2017). Our linear

In this article, we propose a shape optimization algorithm which is able to handle large deformations while maintaining a high level of mesh quality. Based on the method of mappings, we introduce a nonlinear extension operator, which links a boundary control to domain deformations, ensuring admissibility of resulting shapes. The major focus is on comparisons between well-established approaches involving

This work presents an approach to spacecraft attitude motion planning which guarantees rest-to-rest maneuvers while satisfying pointing constraints. Attitude is represented on the group of three dimensional rotations. The angular velocity is expressed as weighted sum of some basis functions, and the weights are obtained by solving a constrained minimization problem in which the objective is the maneuvering

In this paper, we study robust necessary optimality conditions for a nondifferentiable complex fractional programming with uncertain data. A robust counterpart of uncertain complex fractional programming is introduced in the worst-case scenario. The concept of robust optimal solution of the uncertain complex fractional programming is introduced by using robust counterpart. We give an equivalence between

In this paper, we mainly study the Abadie constraint qualification (ACQ) and the strong ACQ of a convex multifunction. To characterize the general difference between strong ACQ and ACQ, we prove that the strong ACQ is essentially equivalent to the ACQ plus the finite distance of two image sets of the tangent derivative multifunction on the sphere and the origin, respectively. This characterization

The aim of this article is to understand how to apply partial or total containment to SIR epidemic model during a given finite time interval in order to minimize the epidemic final size, that is the cumulative number of cases infected during the complete course of an epidemic. The existence and uniqueness of an optimal strategy are proved for this infinite-horizon problem, and a full characterization

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau

We describe an optimization-based tax-aware portfolio construction method that adds tax liability to standard Markowitz-based portfolio construction. Our method produces a trade list that specifies the number of shares to buy of each asset and the number of shares to sell from each tax lot held. To avoid wash sales (in which some realized capital losses are disallowed), we assume that we trade monthly

Decentralized algorithms solve multi-agent problems over a connected network, where the information can only be exchanged with the accessible neighbors. Though there exist several decentralized optimization algorithms, there are still gaps in convergence conditions and rates between decentralized and centralized algorithms. In this paper, we fill some gaps by considering two decentralized algorithms:

Providing the optimal dosing strategy of a drug for an individual patient is an important task in pharmaceutical sciences and daily clinical application. We developed and validated an optimal dosing algorithm (OptiDose) that computes the optimal individualized dosing regimen for pharmacokinetic–pharmacodynamic models in substantially different scenarios with various routes of administration by solving

We investigate when a mean field-type control system can fulfill a given constraint. Namely, given a closed set of probability measures on the torus, starting from any initial probability measure belonging to this set, does there exist a solution to the mean field control system remaining in it for any time? This property—the so-called viability property—is equivalently characterized through a property

In this paper, we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient condition, with or without a quadratic functional growth property. These models include the most well-known classes of objective functions analyzed in the literature: nonsmooth Lipschitz functions and composition of a (potentially) nonsmooth function and

The problem of planar optimal impulsive transfer between ellipses in a Newtonian gravitational field with the final time- free is approached through a transformation of variables. New necessary conditions for optimal Bi-Elliptic Transfers are presented in terms of these transformed variables. The work is applied to examples in which the apses of the ellipses are aligned. The Generalized Hohmann, Bi-Elliptic

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

In this paper, we solve a maximization problem where the objective function is quadratic and convex or concave and the constraints set is the reachable values set of a convergent discrete-time affine system. Moreover, we assume that the matrix defining the system is diagonalizable. The difficulty of the problem lies in the treatment of infinite sequences belonging to the constraint set. Equivalently

In the Target–Attacker–Defender differential game, an Attacker missile strives to capture a Target aircraft. The Target tries to escape the Attacker and is aided by a Defender missile which aims at intercepting the Attacker, before the latter manages to close in on the Target. The conflict between these intelligent adversaries is naturally modeled as a zero-sum differential game. The Game of Degree

The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in finite time.

We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with a decreasing sequence of smoothing

We consider what we term existence-constrained semi-infinite programs. They contain a finite number of (upper-level) variables, a regular objective, and semi-infinite existence constraints. These constraints assert that for all (medial-level) variable values from a set of infinite cardinality, there must exist (lower-level) variable values from a second set that satisfy an inequality. Existence-constrained

This paper aims at solving Zermelo’s navigation problem on conformally flat Riemannian manifolds admitting a ship’s variable self-speed, under the action of arbitrary winds including space and time dependence for both perturbation and ship’s speed. Our approach is a variational one under application of the Euler–Lagrange equations with reference to the initial studies of this problem. First of all

In the development of first-order methods for smooth (resp., composite) convex optimization problems, where smooth functions with Lipschitz continuous gradients are minimized, the gradient (resp., gradient mapping) norm becomes a fundamental optimality measure. Under this measure, a fixed iteration algorithm with the optimal iteration complexity for the smooth case is known, while determining this

The paper is concerned with a kind of minimal time impulse control problem for a semilinear heat equation. We study the existence of optimal controls of this problem, establish a nontrivial Pontryagin’s maximum principle for this problem and then derive the bang–bang property of optimal controls. Based on the existence and the bang–bang property of optimal controls, we discuss the equivalence of the

In this paper, building upon auxiliary principle technique and using proximal operator, we introduce a new explicit algorithm for solving monotone hierarchical equilibrium problems. The considered problem is a monotone equilibrium problem, where the constraint is the solution set of a set-valued variational inequality problem. The strong convergence of the proposed algorithm is studied under strongly

This paper proposes a new algorithm for computing the Nash equilibrium based on an iterative approach of both the proximal and the gradient method for homogeneous, finite, ergodic and controllable Markov chains. We conceptualize the problem as a poly-linear programming problem. Then, we regularize the poly-linear functional employing a regularization approach over the Lagrange functional for ensuring

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

We consider the problem of minimizing a polynomial function of degree three over the boundary of the sphere. If the objective is quadratic instead of cubic, this is the well-studied trust region subproblem, which is known to be tractable. In the cubic case, the problem turns out to be NP-hard. In this paper, we derive and evaluate different approaches for computing lower bounds for the cubic problem

We present an efficient descent method for unconstrained, locally Lipschitz multiobjective optimization problems. The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth multiobjective optimization problems with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy

We present a new theoretical analysis of local superlinear convergence of classical quasi-Newton methods from the convex Broyden class. As a result, we obtain a significant improvement in the currently known estimates of the convergence rates for these methods. In particular, we show that the corresponding rate of the Broyden–Fletcher–Goldfarb–Shanno method depends only on the product of the dimensionality

We consider a mini-batch stochastic Bregman proximal gradient method and a mini-batch stochastic Bregman proximal extragradient method for stochastic convex composite optimization problems. A simplified and unified convergence analysis framework is proposed to obtain almost sure convergence properties and expected convergence rates of the mini-batch stochastic Bregman proximal gradient method and its

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete

We consider an abstract minimization problem in reflexive Banach spaces together with a specific family of approximating sets, constructed by perturbing the cost functional and the set of constraints. For this problem, we state and prove various well-posedness results in the sense of Tykhonov, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, compactness

In this paper, we propose a new viscosity extragradient algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. We prove a strong convergence theorem under some appropriate conditions imposed on the parameters. Finally, we give some numerical experiments to illustrate the advantages of our proposed algorithms. The main results

We study a class of finite horizon optimal economic growth problems with nonlinear utility functions and linear production functions. By using a maximum principle in the optimal control theory and employing the special structure of the problems, we are able to explicitly describe the unique solution via input parameters. Economic interpretations of the obtained results and an open problem about the

A new numerical method is presented for bilevel programs with a nonconvex follower’s problem. The basic idea is to piecewise construct convex relaxations of the follower’s problems, replace the relaxed follower’s problems equivalently by their Karush–Kuhn–Tucker conditions and solve the resulting mathematical programs with equilibrium constraints. The convex relaxations and needed parameters are constructed

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific

We give regularity results for minimizers of two special cases of polyconvex functionals. Under some structural assumptions on the energy density, we prove that minimizers are either bounded, or have suitable integrability properties, by using the classical Stampacchia Lemma.

In this paper, we propose to study a wide class of combinatorial optimization problems called combinatorial optimization problems with labelling. First, we give a combinatorial method to deal with the labelling version of some classical combinatorial optimization problems including minimum vertex cover, maximum independent set, minimum dominating set and minimum set cover, and convert the labelling

This paper expands the existing sufficiency results for the strong minimality of an extremal of the Bolza problem. We cover the case where the strict Legendre Clebsh conditions are not strictly verified. An efficient, easy to use, algorithm to prove minimality is provided. It can be used on solutions with bang-bang control and does not require any local controllability property. The interval where

A new extragradient projection method, which does not require generalized monotonicity, is devised in this paper. In order to ensure its global convergence, we assume only that the Minty variational inequality has a solution. In particular, it applies to quasimonotone variational inequalities having a nontrivial solution.

We propose a nonmonotone trust region method for unconstrained optimization problems on Riemannian manifolds. Global convergence to the first-order stationary points is proved under some reasonable conditions. We also establish local R-linear, super-linear and quadratic convergence rates. Preliminary experiments show that the algorithm is efficient.

We develop an operator splitting approach to solve diffeomorphic matching problems for sequences of surfaces in three-dimensional space. The goal is to smoothly match, at very fast rate, finite sequences of observed 3D-snapshots extracted from movies recording the smooth dynamic deformations of “soft” surfaces. We have implemented our algorithms in a proprietary software installed at The Methodist