Consider the following investment problem over T years, where the objective is to maximize the value...

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Consider the following investment problem over T years, where the objective is to maximize the value of the investments in year T. We assume a perfect capital market with the same annual lending and borrowing rate r> 0 each year. We also assume that exogenous investment funds by are available in year t, for t = 1,..., T. Let n be the number of possible investments. We assume that each investment can be undertaken fractionally (between 0 and 1). Let atj denote the cash flow associated with investment j in year t. Let c; be the value of investment j in year T (including all cash flows subsequent to year T discounted at the interest rate r). The linear program that maximizes the value of the investments in year T is the following. Denote by x, the fraction of investment j undertaken, and let yt be the amount borrowed (if negative) or lent (if positive) in year t. - max 1 Cjxj+YT - Σj=19₁jxj+y₁ ≤ b₁ 1 atjxj - (1+r)yt-1+ y ≤ b 0≤x≤ 1 for t = 2,..., T for j = 1,..., n. (i) Write the dual of the above linear program. (ii) Solve the dual linear program found in (i). [Hint: Note that some of the dual variables can be computed by backward substitution.] (iii) Write the complementary slackness conditions. (iv) Deduce that the first T constraints in the primal linear program hold as equalities. (v) Use the complementary slackness conditions to show that the solution obtained by setting xj = 1 if c; +1(1 + r)T-tatj > 0, and j = 0 otherwise, is an optimal solution. Consider the following investment problem over T years, where the objective is to maximize the value of the investments in year T. We assume a perfect capital market with the same annual lending and borrowing rate r> 0 each year. We also assume that exogenous investment funds by are available in year t, for t = 1,..., T. Let n be the number of possible investments. We assume that each investment can be undertaken fractionally (between 0 and 1). Let atj denote the cash flow associated with investment j in year t. Let c; be the value of investment j in year T (including all cash flows subsequent to year T discounted at the interest rate r). The linear program that maximizes the value of the investments in year T is the following. Denote by x, the fraction of investment j undertaken, and let yt be the amount borrowed (if negative) or lent (if positive) in year t. - max 1 Cjxj+YT - Σj=19₁jxj+y₁ ≤ b₁ 1 atjxj - (1+r)yt-1+ y ≤ b 0≤x≤ 1 for t = 2,..., T for j = 1,..., n. (i) Write the dual of the above linear program. (ii) Solve the dual linear program found in (i). [Hint: Note that some of the dual variables can be computed by backward substitution.] (iii) Write the complementary slackness conditions. (iv) Deduce that the first T constraints in the primal linear program hold as equalities. (v) Use the complementary slackness conditions to show that the solution obtained by setting xj = 1 if c; +1(1 + r)T-tatj > 0, and j = 0 otherwise, is an optimal solution.

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We will first construct the dual linear programme before carrying out the processes as instructed in order to solve the investment problem and provide ... View the full answer