Second-order Sufficient Conditions Research Articles

Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth

In this paper, we study an optimal control problem for a nonlinear system of reaction–diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in H. Garcke, et al. [Math. Model. Methods Appl. Sci. 26 (2016) 1095–1148]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1−norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [P. Colli, et al. To appear in: Appl. Math. Optim. (2019)], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.

Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

The stability of a generalized variational inequality problem with the nuclear norm function

This paper is to study the stability of the generalized variational inequality (GVI) problem whose convex function is the nuclear norm of a linear matrix mapping. The strict Robinson constraint qualification and the second-order sufficient optimality conditions for the GVI problem are proposedand are demonstrated as sufficient conditions for the isolated calmness of the inverse of the KKT (Karush-Kuhn-Tucker) mapping. The constraint non-degeneracy condition and the strong second-order sufficient optimality conditions for the GVI problem are proposed and are proven to be sufficient conditions for the strong regularity of the KKT system.

English

In this paper, we present a sensitivity result for quadratic second-order cone programming under the weak form of second-order sufficient condition. Based on this result, we analyze the local convergence of an SQP-type method for nonlinear second-order cone programming. The subproblems of this method at each iteration are quadratic second-order cone programming problems. Compared with the local convergence analysis done before, we do not need the assumption that the Hessian matrix of the Lagrangian function is positive definite. Besides, the iteration sequence which is proved to be superlinearly convergent does not contain the Lagrangian multiplier.

A Second-Order Sufficient Optimality Condition for Risk-Neutral Bi-level Stochastic Linear Programs

The expectation functionals, which arise in risk-neutral bi-level stochastic linear models with random lower-level right-hand side, are known to be continuously differentiable, if the underlying probability measure has a Lebesgue density. We show that the gradient may fail to be local Lipschitz continuous under this assumption. Our main result provides sufficient conditions for Lipschitz continuity of the gradient of the expectation functional and paves the way for a second-order optimality condition in terms of generalized Hessians. Moreover, we study geometric properties of regions of strong stability and derive representation results, which may facilitate the computation of gradients.

Nonsmooth Vector Optimization Problem Involving Second-Order Semipseudo, Semiquasi Cone-Convex Functions

Recently, Suneja et al. [26] introduced new classes of second-order cone-(η; ξ)-convex functions along with theirgeneralizations and used them to prove second-order Karush–Kuhn–Tucker (KKT) type optimality conditions and duality results for the vector optimization problem involving first-order differentiable and second-order directionally differentiable functions. In this paper, we move one step ahead and study a nonsmooth vector optimization problem wherein the functions involved are first and second-order directionally differentiable. We introduce new classes of nonsmooth second-order cone-semipseudoconvex and nonsmooth second-order cone-semiquasiconvex functions in terms of second-order directional derivatives. Second-order KKT type sufficient optimality conditions and duality results for the same problem are proved using these functions.

The Geometric Properties of a Class of Nonsymmetric Cones

Geometric methods are important for researching the differential properties of metric projectors, sensitivity analysis, and the augmented Lagrangian algorithm. Sun [3] researched the relationship among the strong second-order sufficient condition, constraint nondegeneracy, B-subdifferential nonsingularity of the KKT system, and the strong regularity of KKT points in investigating nonlinear semidefinite programming problems. Geometric properties of cones are necessary in studying second-order sufficient condition and constraint nondegeneracy. In this paper, we study the geometric properties of a class of nonsymmetric cones, which is widely applied in optimization problems subjected to the epigraph of vector k-norm functions and low-rank-matrix approximations. We compute the polar, the tangent cone, the linear space of the tangent cone, the critical cone, and the affine hull of this critical cone. This paper will support future research into the sensitivity and algorithms of related optimization problems.

Augmented Lagrangians quadratic growth and second-order sufficient optimality conditions

It is well-known that the primal quadratic growth condition of the classical augmented Lagrangian around a local minimizer can be obtained under the second-order sufficient optimality condition. In this paper, we show that those conditions are indeed equivalent. Moreover, we prove that the primal quadratic growth condition of the sharp augmented Lagrangian around a local minimizer is in fact equivalent to the weak second-order sufficient optimality condition. In addition, we present some secondary results involving the sharp augmented Lagrangian.

A Neural Network Based on the Metric Projector for Solving SOCCVI Problem.

We propose an efficient neural network for solving the second-order cone constrained variational inequality (SOCCVI). The network is constructed using the Karush-Kuhn-Tucker (KKT) conditions of the variational inequality (VI), which is used to recast the SOCCVI as a system of equations by using a smoothing function for the metric projection mapping to deal with the complementarity condition. Aside from standard stability results, we explore second-order sufficient conditions to obtain exponential stability. Especially, we prove the nonsingularity of the Jacobian of the KKT system based on the second-order sufficient condition and constraint nondegeneracy. Finally, we present some numerical experiments, illustrating the efficiency of the neural network in solving SOCCVI problems. Our numerical simulations reveal that, in general, the new neural network is more dominant than all other neural networks in the SOCCVI literature in terms of stability and convergence rates of trajectories to SOCCVI solution.

A Study of Piecewise Linear-Quadratic Programs

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 (PLQ) objective functions. Starting from a study of the representation of such a function in terms of a family of elementary functions consisting of squared affine functions, squared plus-composite-affine functions, and affine functions themselves, we summarize some local properties of a PLQ function in terms of their first and second-order directional derivatives. We extend some well-known necessary and sufficient second-order conditions for local optimality of a quadratic program to a PLQ program and provide a dozen such equivalent conditions for strong, strict, and isolated local optimality, showing in particular that a PLQ 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 PLQ program is finite; this result supplements a recent result about the finite number of directional stationary objective values. Interestingly, these finiteness results can be uncovered by invoking a very powerful property of subanalytic functions; our proof is fairly elementary, however. We discuss applications of PLQ programs in some modern statistical estimation problems. These problems lead to a special class of unconstrained composite programs involving the non-differentiable $\ell_1$-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.

Second-Order Optimality Conditions in Locally Lipschitz Inequality-Constrained Multiobjective Optimization.

The main goal of this paper is to give some primal and dual Karush–Kuhn–Tucker second-order necessary conditions for the existence of a strict local Pareto minimum of order two for an inequality-constrained multiobjective optimization problem. Dual Karush–Kuhn–Tucker second-order sufficient conditions are provided too. We suppose that the objective function and the active inequality constraints are only locally Lipschitz in the primal necessary conditions and only strictly differentiable in sense of Clarke at the extremum point in the dual conditions. Examples illustrate the applicability of the obtained results.

Strong Local Optimality for a Bang–Bang–Singular Extremal: General Constraints

The paper provides second-order sufficient conditions for the strong local optimality of bang–bang–singular extremals in a Mayer problem with general end point constraints. The sufficient conditions are expressed as a strengthening of the necessary ones plus the coerciveness of a suitable quadratic form related to a sub-problem of the given one. The sufficiency of the given conditions is proven via Hamiltonian methods.

A regularization method for constrained nonlinear least squares

We propose a regularization method for nonlinear least-squares problems with equality constraints. Our approach is modeled after those of Arreckx and Orban (SIAM J Optim 28(2):1613–1639, 2018. https://doi.org/10.1137/16M1088570) and Dehghani et al. (INFOR Inf Syst Oper Res, 2019. https://doi.org/10.1080/03155986.2018.1559428) and applies a selective regularization scheme that may be viewed as a reformulation of an augmented Lagrangian. Our formulation avoids the occurrence of the operator $$A(x)^T A(x)$$, where A is the Jacobian of the nonlinear residual, which typically contributes to the density and ill conditioning of subproblems. Under boundedness of the derivatives, we establish global convergence to a KKT point or a stationary point of an infeasibility measure. If second derivatives are Lipschitz continuous and a second-order sufficient condition is satisfied, we establish superlinear convergence without requiring a constraint qualification to hold. The convergence rate is determined by a Dennis–More-type condition. We describe our implementation in the Julia language, which supports multiple floating-point systems. We illustrate a simple progressive scheme to obtain solutions in quadruple precision. Because our approach is similar to applying an SQP method with an exact merit function on a related problem, we show that our implementation compares favorably to IPOPT in IEEE double precision.

The rate of convergence of proximal method of multipliers for second-order cone optimization problems

In this paper we consider a proximal method of multipliers (PMM) for a nonlinear second-order cone optimization problem. With the assumptions of constraint nondegeneracy, strict complementarity and second-order sufficient condition, we estimate the local convergence rate of PMM to be linear or superlinear, which depends on the strategy of parameter selection.

Variable-time-domain neighboring optimal guidance and attitude control of low-thrust lunar orbit transfers

Lunar orbit dynamics and transfers at low altitudes are subject to considerable perturbations related to the gravitational harmonics associated with the irregular lunar mass distribution. This research proposes the combination of two techniques applied to low-thrust lunar orbit transfers, i.e. (i) the variable-time-domain neighboring optimal guidance (VTD-NOG), and (ii) a proportional-derivative attitude control algorithm based on rotation matrices (PD-RM). VTD-NOG belongs to the class of feedback implicit guidance approaches, aimed at maintaining the spacecraft sufficiently close to the reference trajectory. This is an optimal path that satisfies the second-order sufficient conditions for optimality. A fundamental original feature of VTD-NOG is the use of a normalized time scale, with the favorable consequence that the gain matrices remain finite for the entire time of flight. VTD-NOG identifies the trajectory corrections by assuming the thrust direction as the control input. Because the thrust direction is fixed with respect to the spacecraft, VTD-NOG generates the desired orientation pursued by the attitude control system. A proportional-derivative approach using rotation matrices (PD-RM) is employed in order to drive the actual spacecraft orientation toward the desired one. Reaction wheels are considered as the actuators that perform attitude control. Extensive Monte Carlo simulations are performed, in the presence of nonnominal flight conditions related to (i) lunar gravitational harmonics, (ii) gravitational pull of the Earth and the Sun as third bodies, (iii) unpredictable propulsive fluctuations, and (iv) errors on initial attitude. The numerical results unequivocally demonstrate that the joint use of VTD-NOG and PD-RM control represents an accurate and effective methodology for guidance and control of low-thrust lunar orbit transfers.

The combined effect of optimal control and swarm intelligence on optimization of cancer chemotherapy

Background and Objectives: In cancer therapy optimization, an optimal amount of drug is determined to not only reduce the tumor size but also to maintain the level of chemo toxicity in the patient's body. The increase in the number of objectives and constraints further burdens the optimization problem. The objective of the present work is to solve a Constrained Multi- Objective Optimization Problem (CMOOP) of the Cancer-Chemotherapy. This optimization results in optimal drug schedule through the minimization of the tumor size and the drug concentration by ensuring the patient's health level during dosing within an acceptable level.Methods: This paper presents two hybrid methodologies that combines optimal control theory with multi-objective swarm and evolutionary algorithms and compares the performance of these methodologies with multi-objective swarm intelligence algorithms such as MOEAD, MODE, MOPSO and M-MOPSO. The hybrid and conventional methodologies are compared by addressing CMOOP.Results: The minimized tumor and drug concentration results obtained by the hybrid methodologies demonstrate that they are not only superior to pure swarm intelligence or evolutionary algorithm methodologies but also consumes far less computational time. Further, Second Order Sufficient Condition (SSC) is also used to verify and validate the optimality condition of the constrained multi-objective problem.Conclusion: The proposed methodologies reduce chemo-medicine administration while maintaining effective tumor killing. This will be helpful for oncologist to discover and find the optimum dose schedule of the chemotherapy that reduces the tumor cells while maintaining the patients’ health at a safe level.

Joint Power Allocation for NOMA-Based Diamond Relay Networks With and Without Cooperation

Non-orthogonal multiple access (NOMA)-based diamond relaying (NDR) is an efficient approach for combining NOMA and relaying techniques in such a way as to enhance the achievable rate from the source to the destination in the network. This paper examines the problem of joint power allocation among all the transmission phases during the operation of such networks. Based on the Karush-Kuhn-Tucker (KKT) condition and the second-order sufficient condition (SOSC), at least the local optimal solution is derived and analyzed. In addition, a new protocol for NDR networks based on cooperative communications is introduced and the associated joint power-allocation problem is examined. It is shown that this cooperative NDR (C-NDR) network further improves the achievable rate in some typical placements of the relays. Simulation results verify the correctness of the deviation and confirm the effectiveness of the proposed joint power-allocation method for both NDR and C-NDR networks.

On Optimal Control Problems with Controls Appearing Nonlinearly in an Elliptic State Equation

An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection technique. First-order necessary optimality conditions are derived and two types of second-order sufficient optimality conditions are established. A first theorem invokes a well-known assumption on the set of zeros of the switching function. A second relies on coercivity of the second derivative of the reduced objective functional. The results are applied to the convergence of optimal state functions for a finite element discretizion of the control problem.

An optimal control problem of the 3D viscous Camassa–Holm equations

In this paper, we study an optimal control problem of the three-dimensional viscous Camassa–Holm equations in bounded domains with Dirichlet boundary conditions and a quadratic objective functional. We first prove the existence of an optimal solution. Then we establish the first-order necessary and second-order sufficient optimality conditions.