A generalized projection-based scheme for solving convex constrained optimization problems

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.

Purpose: To support IMRT research, we developed an open‐source, computationally efficient, and flexible optimization solver that can solve a wide range of problem formulations, including constrained optimization, convex and non‐convex polynomial functions, and hierarchical optimization formulations. Methods: The solver is an adaptation and modification of the open‐source Interior Point Optimization Solver (IPOPT), with some performance enhancements. Previously, we implemented the IMRT optimization based on the commercial Mosek solver. Mosek can solve convex linear and quadratic problems but cannot solve convex or non‐convex polynomials of higher orders. This limits the ability to model IMRT optimization using more appropriate metrics like gEUD. Therefore, we implemented the optimization problem using C++ interface of IPOPT. IPOPT can use the Newton or Quasi‐Newton method to solve the optimization problem. In the Newton method, the exact Hessian matrix provided by the user is used to find the direction in each iteration while in the Quasi‐Newton method the Hessian matrix is approximated from the Jacobian matrix. After integrating IPOPT with CERR, we found that the Newton method converges more quickly than the Quasi‐Newton method. To increase the computational efficiency, we incorporated the uBLAS matrix library. Results: Few compared the modified IPOPT code to Mosek for a set of convex quadratic problems using realistic, anonymized treatment planning data and dose data accessed using CERR. These problems have a unique global minimum and both solvers were observed to give the same results for eight head and neck planning datasets. The median speedup of IPOPT compared to Mosek was a factor of 3.5. Conclusion: Our IPOPT implementation provides an open‐source framework to model the IMRT optimization using complex objective functions; is integrated with CERR; and our C++ implementation of the optimization problem gives increased control over the optimization process. This tool provides a basis for future IMRT treatment planning research. Memorial Sloan‐Kettering Cancer Center

Inertial primal-dual projection neurodynamic approaches for constrained convex optimization problems and application to sparse recovery.

We consider bilevel problem in the following formulation. The follower problem is a convex quadratic problem which linearly depends on the leader variable. The leader problem is a quadratic problem. We are looking for an optimistic solution. Optimal value function of the follower problem is used to reformulate bilevel problem as a standard (one-level) optimization problem. By this way we obtain a nonconvex multiextremal problem with implicit nonconvex constraint generated by the optimal value function. We show how to construct an explicit piecewise quadratic function which is support to the optimal value function at a given point. Usage of the support functions allows us to approximate the implicit one-level problem by a number of explicit nonconvex quadratic problems. In our talk we describe an iterative procedure based on such approximation.

Optimization problems are used to model many real-life problems. Therefore, solving these problems is one of the most important goals of algorithm design. A general optimization problem can be defined by specifying a set of constraints that defines a subset in some underlying space (like the Euclidean space) called the feasible subset and an objective function that we are trying to maximize or minimize, as the case may be, over the feasible set. The difficulty of solving such problems typically depends on how ‘complex’ the feasible set and the objective function are. For example, a very important class of optimization problems is linear programming. Here the feasible subset is specified by a set of linear inequalities (in the Euclidean space); the objective function is also linear. A more general class of optimization problems is convex programming, where the feasible set is a convex subset of a Euclidean space and the objective function is also convex. Convex programs (and hence, linear programs) have a nice property that any local optimum is also a global optimum for the objective function. There are a variety of techniques for solving such problems – all of them try to approach a local optimum (which we know would be a global optimum as well). These notions are discussed in greater detail in a later section in this chapter. The more general problem, the so-called non-convex programs, where the objective function and the feasible subset could be arbitrary can be very challenging to solve. In particular, discrete optimization problems, where the feasible subset could be a (large) discrete subset of points falls under this category.

In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems.

We consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities. At each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut (defined by a convex quadratic inequality) is placednear the approximate center. As the number of cuts increases, the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem. Previously, similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem. Our method differs from these early methods in that we use "quadratic cuts" in the computation instead of linear cuts. Moreover, our method has a polynomial worst case complexity of $O(n\ln \frac{1}{\varepsilon})$ on the total number of cuts to be used, where n is the number of convex quadratic polynomial inequalities in the problem and $\varepsilon$ is the radius of the largest ball contained in the feasible set. In contrast, the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time.

IntroductionSystematic reviews demonstrate that advance care planning (ACP) has many positive effects for residents of aged care facilities, including decreased hospitalisation. The proposed Residential Aged Care Facility (RACF) ‘Goals of...

The rise of young-onset colorectal cancer (YO-CRC) is an alarming public health issue. The proportion of new cases diagnosed in young people (20-49) had increased from 6% in 1990 to 11% in 2013 and coincided with the declining CRC cases in older people. YO-CRC patients face unique clinical challenges, as many are diagnosed at advanced stages of the disease and subjected to aggressive treatments. With the increase of targeted therapies and precision medicine available to them, patients.and their caregivers must understand the different treatment plans and options that could be suitable for them. So the decisions are best aligned with their unique tumor characteristics and the preferences of patients and their caregivers and articulated appropriately to providers. A cross-sectional study was conducted and patients and survivors (N=885) completed an online survey with 11 questions related to biomarkers and treatment options literacy. In addition, self-identified unrelated caregivers of YO-CRC patients (N=203) completed a similar survey. The average age of patients, survivors, and caregivers was 40-49. The majority of patients/survivors are college graduates (73%) and (40%) completed an advanced degree. Similarly, caregivers were mostly college graduates (79%) and (43%) had an advanced degree. Despite the fact that 73% of stage IV patients felt informed before treatment began, only 54% reported they received biomarker testing before treatment was initiated. However, out of these respondents (41%) stated their provider discussed biomarker testing at diagnosis and only half were informed about their results. The majority of respondents stated they were not made aware of biomarkers from their medical team and their patient biomarker education came from an internet search (56%), a family friend (22%) or patient navigator (22%). A majority were female (91%) were more likely to have biomarkers tested than males (9%) with 60% living in a suburban area, 20% urban area and in 18% in rural areas. While many caregivers report they serve as a liaison between patients and medical information retrieval, the majority of caregivers (55%) reported that they did not fully understand the critical aspects of patients' medical treatment plans and needs. Multiple studies have shown that patient-centered care improves patients' outcomes. Our organization promotes patient empowerment and encourages patients and their families to use medical information to develop an evidence-based treatment plan together with their physicians and care team to provide a roadmap for patients to navigate their journey. Our study identified numerous obstacles to achieving this goal. Further studies of physicians and other medical team members may shed light on reasons why the majority of patients and their caregivers reported a gap in information and understanding medical needs. Citation Format: Kimberly Newcomer, Tamara Springer, Never2Young Board, Danielle Peterson, Ronit Yarden. Gaps in biomarkers and health literacy among young-onset colorectal cancer patients, survivors and caregivers [abstract]. In: Proceedings of the Annual Meeting of the American Association for Cancer Research 2020; 2020 Apr 27-28 and Jun 22-24. Philadelphia (PA): AACR; Cancer Res 2020;80(16 Suppl):Abstract nr 5738.

e16115 Background: The proportion of new young-onset colorectal cancer (YO-CRC) cases diagnosed in young people (20-49) had doubled over the past 3 decades, and it is becoming an alarming public health issue. YO-CRC patients face unique clinical challenges, as many are diagnosed at advanced stages of the disease and subjected to aggressive treatments. With the increase of targeted therapies and precision medicine available, NCCN and ASCO guidelines recommend that all stage IV, metastatic CRC patients be tested for 4 biomarkers in addition to MSI-H. However, not all eligible patients benefit from biomarker testing and biomarker-driven therapies. Methods: We conducted an online survey to assess patients, survivors and caregivers’ knowledge and understanding of biomarker testing and whether they feel empowered to discuss with their doctor different treatment options based on their unique tumor characteristics and their preferences. Results: The cross-sectional study was completed by patients and survivors (N = 885) and unrelated caregivers of YO-CRC patients (N = 203). The median age of patients and survivors was 42, and the median age of caregivers was also 42. The majority of the participants were college graduates. Although 82% of stage IV, YO-CRC patients felt informed before treatment began, only 70% indicated their tumor was tested for biomarkers, and 54% of them received biomarker testing before treatment initiation. Only 33% of patients reported they became aware of biomarkers by their medical team while others indicated resources such as internet searches, family and friends, or patient navigators. Many of the caregivers who reported that they served as a liaison between patients and medical information retrieval felt that they did not fully understand the critical aspects of the patients' medical treatment plan. Conclusions: Multiple studies have shown that patient-centered care improves patients’ outcomes. Our organization encourages patients and their families to use medical information to navigate their journey. Our patient-reported study suggests that there are gaps in patients’ and caregiver’s understanding of biomarker testing that may hinder patient's empowerment for reaching evidence-based shared-decision treatment plan together with their physicians.

Tree-seed algorithm (TSA) is a nature-inspired and population-based algorithm for solving continuous optimization problems. The tree-seed relationship is the main motivation of this algorithm. TSA has only two peculiar parameters which are the total number of trees in the stand (pop) and the controller of the seed production (search tendency). Although many problems have been solved in the literature by TSA which is the successful optimizer for low dimensional unconstrained continuous problems, real-life problems have not been addressed yet. In this study, six continuous unconstrained real-life optimization problems (gas transmission compressor design, optimal capacity of gas production facility, gear train design, frequency modulation sounds parameter identification, the spread spectrum radar polyphase code design with 10 decision variables, the spread spectrum radar polyphase code design with 20 decision variables) have been solved. It is seen that choosing the number of the population as 50 and the value of search tendency as 0.1 is appropriate according to experimental results for these problems.

The distributed time‐varying constrained convex optimization problem over a connected undirected network topology is studied in this article. The proposed distributed optimization algorithm is composed of a projection map term, a consensus term, and a gradient term, and is feasible for any initial condition. We first show that the states of the optimization algorithm uniformly converge to a neighborhood of the constraint set while an approximate global consensus is achieved. Then the states are proven to uniformly converge to a neighborhood of the time‐varying optimal trajectory. The theoretical results are validated by some simulations.

The prediction of clinical outcome of patients after breast cancer surgery plays an important role in medical tasks like diagnosis and treatment planning. These kinds of estimations are currently performed by clinicians using non-numerical techniques. Artificial neural networks are shown to be a powerful tool for analyse data sets where there are complicated non-linear interactions between the input data and the information to be predicted, and fuzzy logic appears as an useful tool to perform decision making in real life problems. In this paper, we present an hybrid neuro-fuzzy prognosis system for the prediction of patients relapse probability using clinical-pathological data (tumor size, patient age, estrogens receptors, etc.) from the Medical Oncology Service of the Hospital Clinical University of Malaga. Results show the classification accuracy improvement obtained by the proposed model in comparison with an approach based on exclusively on artificial neural networks proposed in our previous work.

We study the convergence behavior of a class of projection methods for solving convex feasibility and optimization problems. We prove that the algorithms in this class converge to solutions of the consistent convex feasibility problem, and that their convergence is stable under summable perturbations. Our class is a subset of the class of string-averaging projection methods, large enough to contain, among many other procedures, a version of the Cimmino algorithm, as well as the cyclic projection method. A variant of our approach is proposed to approximate the minimum of a convex functional subject to convex constraints. This variant is illustrated on a problem in image processing: namely, for optimization in tomography.

Multilevel optimization problems deals with mathematical programming problems whose feasible set is implicitly determined by a sequence of nested optimization problems. These kind of problems are common in different applications where there is a hierarchy of decision makers exists. Solving such problems has been a challenge especially when they are non linear and non convex. In this paper we introduce a new algorithm, inspired by natural adaptation, using (1+1)-evolutionary strategy iteratively. Suppose there are k level optimization problem. First, the leader’s level will be solved alone for all the variables under all the constraint set. Then that solution will adapt itself according to the objective function in each level going through all the levels down. When a particular level’s optimization problem is solved the solution will be adapted the level’s variable while the other variables remain being a fixed parameter. This updating process of the solution continues until a stopping criterion is met. Bilevel and trilevel optimization problems are used to show how the algorithm works. From the simulation result on the two problems, it is shown that it is promising to uses the proposed metaheuristic algorithm in solving multilevel optimization problems.KeywordsMultilevel optimization(1+1)-Evolutionary strategymetaheuristic algorithmsNatural adaptation