 # Stock Portfolio Optimization

provides functions for minimizing objective functions, possibly subject to constraints. It includes solvers for nonlinear problems , linear programing, constrained and nonlinear least-squares, root finding, and curve fitting. In order to converge more quickly to the solution, this routine uses the gradient of the objective function. If the gradient is not given by the user, then it is estimated using first-differences. The Broyden-Fletcher-Goldfarb-Shanno method typically requires fewer function calls than the simplex algorithm even when the gradient must be estimated. Note that the Rosenbrock function and its derivatives are included inscipy.optimize.

### What does Scipy optimize minimize?

scipy provides scipy.optimize.minimize() to find the minimum of scalar functions of one or more variables. The simple conjugate gradient method can be used by setting the parameter method to CG.

We look forward to sharing our expertise, consulting you about your product idea, or helping you find the right solution for an existing project. Raw materials are brought to the first plant from the first warehouse and from the third warehouse . Raw materials are brought to the second plant from the second warehouse and from the third warehouse . In total, both plants will receive 8 tons of raw materials, as required at the lowest possible cost. That is, raw materials from the warehouse 3 and warehouse 1 for 2 and 6 tons, respectively, are brought to the first plant. And raw materials from a warehouse of 2 and warehouse 1 of 4 tons are brought to the second plant. That is, each plant will receive 8 tons of raw materials, as was necessary.

I hope this has provided a slightly clearer explanation for you. These optimization algorithms can be used directly in a standalone manner to optimize a function. Most notably, algorithms for local search and algorithms for global search, the two main types of optimization you may encounter on a machine learning project. Method trust-ncg uses the Newton conjugate gradient trust-region algorithm for unconstrained minimization. This algorithm requires the gradient and either the Hessian or a function that computes the product of the Hessian with a given vector. It differs from the Newton-CGmethod described above as it wraps a C implementation and allows each variable to be given upper and lower bounds. Method trust-krylov uses the Newton GLTR trust-region algorithm , for unconstrained minimization. On indefinite problems it requires usually less iterations than thetrust-ncg method and is recommended for medium and large-scale problems. The simplex algorithm is probably the simplest way to minimize a fairly well-behaved function. It requires only function evaluations and is a good choice for simple minimization problems. However, because it does not use any gradient evaluations, it may take longer to find the minimum.

## Bounded Minimization (method=’bounded’)¶

It performs sequential one-dimensional minimizations along each vector of the directions set , which is updated at each iteration of the main minimization loop. The function need not be differentiable, and no derivatives are taken. If bounds are not provided, then an unbounded line search will be used. If bounds are provided and the initial guess is within the bounds, then every function evaluation throughout the minimization procedure will be within the bounds. If direc is not full rank, then some parameters may not be optimized and the solution is not guaranteed to be within the bounds. Scipy.optimize.linprog is the Python library to minimize a linear objective function subject to linear equality and inequality constraints. Method COBYLA uses the Constrained Optimization BY Linear Approximation method, , . For nonlinear least squares with bounds on the variables use least_squares in scipy.optimize. Finding a root of a set of non-linear equations can be achieved using the root() function.

## 7 1.3. Noisy Versus Exact Cost Functions¶

If you are a BYU student and want access to one of the optimizers in bold, please come see me. Please note that the optimizers SNOPT, NLPQLP, and our version of IPOPT are for academic use only, and you should cite the authors if used for any publications. fmincon, gradient-based, nonlinear constrained, includes an interior-point, sqp, active-set, and trust-region-reflective scipy optimize constraints method. fminunc, gradient-based, nonlinear unconstrained, includes a quasi-newton and a trust-region method. In general the Simplex solver from scipy.optimize.linprog should be avoided. The new interior-point solver available in SciPy 1.0 seems to behave much better. Every single day there is new information that you could use to recalucate the weights for your portfolio.

In the function definition, you can use any mathematical functions you want. The only limit is that the function must return a single number at the end. In this code, the first line finds the code associated with ham messages. According to our hypothesis above, the ham messages have the fewest digits, and the digit array was sorted from fewest to most digits.

## Linear Algebra¶

SciPy optimize package provides a number of functions for optimization and nonlinear equations solving. One such function offshore mobile application development is minimize which provides a unified access to the many optimization packages available through scipy.optimize.

If None or False, the gradient will be estimated using 2-point finite difference estimation with an absolute step size. Alternatively, the keywords can be used to select a finite difference scheme for numerical estimation of the gradient with a relative step size. These finite difference schemes obey any specified bounds. That is because the conjugate gradient algorithm approximately solve the trust-region subproblem by iterations without the explicit Hessian factorization.

Unfortunately my code that throws this error with 1.5.0 but not with 1.4.1 is a bit long and relies on data from an outside file, so I can’t really post it here as an example. The example below demonstrates how to solve a two-dimensional multimodal function using simulated annealing. The function that is being optimized is typically nonlinear, nonconvex, and may have one or more than one input variable. Many of the algorithms are used as building blocks for other algorithms within the SciPy library, as well as machine learning libraries such as scikit-learn. The Python SciPy open-source library for scientific computing provides a suite of optimization techniques.

## Solution 4: Eliminate T Get Single Constraint¶

Here we define a list of a bunch of stock tickers to pick from to put in our portfolio. In general for linear and integer programs, I recommend other specialized solvers like CPLEX and Gurobi which all have python interfaces. For other open source solvers which interface with python see here. If you want to use Conjugate Gradient, then you can just change the method parameter. When an optimum exists for the unconstrained problem (e.g. with an infinite budget), it is called a bliss point, or satiation. This introduces a python application to help configure your model. A full optimization of a AMS model, and a easy template to build other and more advanced optimization processes upon.

Outside of SciPy you can also consider cvxopt package by S. Vandenberghe, the authors of the book Convex Optimization. This package can do much more than just linear programming and has its own sparse matrix formats. It can be done using stages of the systems development life cycle the nonlinear least squares function scipy.optimize.leastsq. The parameters thus discovered are used to draw the signal as the sum of the discovered Gaussians below. Solve a nonlinear least-squares problem with bounds on the variables.

## 5 5. Optimization And Fit: Scipy.optimize¶

You need to count the number of digits that appear in each text message. Python includes collections.Counter in the standard library to collect counts of objects in a dictionary-like structure. However, since all of the functions in scipy.cluster.vq expect NumPy arrays as input, you can’t use collections.Counter for this example. Instead, you use a NumPy array and implement scipy optimize constraints the counts manually. Clustering is a popular technique to categorize data by associating it into groups. The SciPy library includes an implementation of the k-means clustering algorithm as well as several hierarchical clustering algorithms. In this example, you’ll be using the k-means algorithm in scipy.cluster.vq, where vq stands for vector quantization.

### How do you find optimal profit?

The key goal for a perfectly competitive firm in maximizing its profits is to calculate the optimal level of output at which its Marginal Cost (MC) = Market Price (P). As shown in the graph above, the profit maximization point is where MC intersects with MR or P.

The optimisation framework provides two ways how this can be achieved. In the above chart we can see the efficient frontier denoted by ‘x’s’. The big red star is the portfolio optimized for Sharpe Ratio, and the Yellow star is the portfolio scipy optimize constraints is optimized to minimize variance . This tells us that a portfolio of 45.69% TLT, 15.07% GLD, and 39.24% QQQ will give us the best risk adjusted returns. We can pull out the individual performance parameters of this portfolio accordingly.

Each optimisation algorithm supports different features and hence has different configuration options. To be able to access these options, any arguments that are unknown to minimize or maximize will be passed to the optimisation algorithm. The efficient frontier is defined as all the portfolios that maximize the return for a given level of volatility. There can only be one of these for each level of volatility, and when plotted forms a curve around the cluster of portfolio values. The next thing we do is calculate the portfolio variance by way of the following.

Appreciate if you can instruct on how to adjust the parameter or shall we use other Python package to solve the problem. @Yufeng.Ling FYI I have just written a Jupyter notebook that goes through different portfolio optimisation methods using a recent library called mlfinlab. I am a full-time consultant and provide services related to the design, implementation and deployment of mathematical programming, cloud computing definition optimization and data-science applications. Usually I cannot blog about projects I am doing, but there are many technical notes I’d like to share. Not in the least so I have an easy way to search and find them again myself. Scipy also contains other root finding techniques such as false position, Brent, and Newton’s method. If you want to solve, a systems of linear equations, use the solve function.